My Airbag indicator is on steady on my 2004 Ford Explorer

you can reset the code with a code reader, or the FIXD App Module. I have one

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Airbag Light Codes for Fords - It Still Runs

https://itstillruns.com > airbag-light-codes-fords-7652177
When there is a problem with a Ford's airbag, the airbag warning light will illuminate and flash a code that identifies the problem. Because airbag problems ...

53 Airbag Code - 53 SRS Light For Ford Falcon

https://airbagcode.com > ford-falcon > 53
53 airbag error code for Ford Falcon and possible reason of 53 srs is Air Bag Tone Warning Indicator Circuit Failure.
Missing: 1996 ‎' Must include: 1996

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  53 Airbag Code - 53 SRS Light For Ford F - AirbagCode.com

  https://airbagcode.com > ford-f > 53
  53 airbag error code for Ford F and possible reason of 53 srs is Air Bag Tone Warning Indicator Circuit Failure.

Key switches can do some funny symptoms so I'd giggle that good for a beginning and then check the schematics from that location out.

Have a dealer or repair shop scan the truck
for body codes

They will be enhanced & not avail for free or with
a code reader

Need factory reference material to look them up &
follow diagnostic charts to resolve

Replace the airbag lamp in the cluster, and get the LFC ( Light Flash Code ) to figure out what the problem is with the SRS.
The 5x5 beeping is an indication that there is a SRS problem waiting and the airbag light cannot give the LFC. Page 105 in the owners manual.

Once you replace the airbag light, you will get a LFC that is a 2 digit number presented by the flashes.
Example is a LFC of 42, the airbag light would flash 4 times, pause flash 2 times pause and repeat.

Once you have the LFC, either start a new problem, or add a comment to this problem with the number for further diagnoses.

upon ignition on the airbag diagnostic module will scan the system for any fault, if there is a fault the airbag indicator will flash. The first set of flash signifies the first digit then a pause the next set of flash for the second digit.Combine these two digits denotes the fault code i.e (two rapid flashes means 2 then a pause followed by three flashes means 3 combine these will read 23 this is the fault code but you need to consult this in the repair manual for ford cars to get the particular fault.Repetitions of 5 sets will occur then steady light on for the airbag.Just replace the instrument cluster and replace the busted bulbs for odometer and turn indicator

When you turn on the ignition switch does the light flash and then stay steady on?

If it's flashing it could indicate the fault code. By instance, two long followed by 5 short blinking will indicate an error code 25

The same code may be repeated before the light stay on.

By the way, any changes or repairs done recently on your vehicule?
Let us know if you get additional information
Hope it will help
Dan

Common fault on FORD
Usually due to a poor connection in the YELLOW PLASTIC CONNECTOR underneath the passenger seat.
Just push it together properly and it should clear the AirBag fault

Safety Belt pre-tensioner Status (Driver pre-tensioner Circuit Fault).
On the seat belt buckle there is a pre-tensioner that when vehicle is involved in an accident it will deploy and pull the belt tighter.

If the vehicle has never been involved in an accident it could be what we call terminal fretting which is just a build up on the terminals which cause a open circuit. Usually just unplugging and reconnecting the connector a couple of times will fix this.

If this doesn't correct it I would recommend having your local dealer look at it.

it resets itself.

Principles of Operation
The PATS uses a specially encoded ignition key. Each encoded ignition key contains a permanently installed electronic device called a transponder. Each transponder contains a unique electronic identification code out of over 18 billion, billion combinations.
The passive anti-theft system (PATS), also known as SecuriLock ®, uses radio frequency identification technology to deter a drive-away theft. This system is known as SecuriLock® in North America, Safeguard® in the U.K., and PATS in Continental Europe. Passive means that it does not require any activity from the user.
The SecuriLock® System (PATS) is not compatible with aftermarket remote start systems, which allow the vehicle to be started from outside the vehicle. These systems may reduce the vehicle security level, and also may cause no-start issues. If equipped the remote start system must be removed before investigation of PATS-related, no-start issues.
Each encoded ignition key must be programmed into the vehicle's powertrain control (PCM) before it can be used to start the engine. There are special diagnostic repair procedures described in this section that must be carried out if a new encoded ignition key is necessary.
This system contains a new feature named Unlimited Key Mode. This feature allows a customer to program more than eight keys to the vehicle if they request it. Each vehicle in Unlimited Key Mode is set up with a special Unlimited Transponder Security Key. This allows all the customer vehicles to share the same keys, but no other keys from outside can be used to operate the vehicles. For an individual customer, any randomly selected Security Key is acceptable. Refer to Unlimited Key Mode Programming in Key Programming Switch State Control in this section.
The PATS transceiver module communicates with the encoded ignition key. The module is located behind the steering column shroud and contains an antenna connected to a small electronics module. During each vehicle start sequence, the transceiver module reads the encoded ignition key identification code and sends data to the PCM.
The control functions are contained in the PCM. This module carries out all of the PATS functions, such as receiving the identification code from the encoded ignition key and controlling the engine enable. The PCM initiates the key interrogation sequence when the vehicle ignition switch is turned to RUN or START.
All elements of the PATS must be functional before the engine is allowed to start. If any of the components are not working correctly, the vehicle will not start.
The PATS uses a visual theft indicator. The indicator will prove out for three seconds when the ignition switch is turned to RUN or START under normal operation. If there is a PATS concern, this indicator will either flash rapidly or glow steadily when the ignition switch is turned to RUN or START. The PATS system also flashes the theft indicator every two seconds at ignition OFF to act as a visual deterrent.
The PATS will be activated and will disable the vehicle from starting if there is a:

• incorrectly encoded ignition key
• damaged encoded ignition key
• unprogrammed key
• non-encoded key (key has no electronics)
• damaged wiring
• damaged transceiver
• damaged PCM

Sounds like a connection problem/error. When you replaced the starter solenoid, do you remember how many wires you disconnected and reconnected?

Was there any indication ( sparks, etc) of accidentally causing a short when you replaced the solenoid?

Your initial problem of no start but hearing the click sound is almost a sure fire indication that the starter is NG. If the solenoid clicks, that means that it is applying 12 volts to the starter.

hope this helps Good luck Loringh