Should I paint or stain the old railroad ties on the wall in my patio?

You can use a clear stain as a preservative. Paint will eventually peel off.

Have you tried power washing them? I would start with that and see how they look and if you want to still add color, I would stain. The problem with doing a stain or paint, they will need to be maintained every few years.

If they're true repurposed railway ties, be very careful as they will have been treated with creosote. It's unlikely any kind of paint or stain will adhere to them. Power washing may remove the surface creosote, but it's hard to say if any kind of finish will adhere even after the washing.

I have been a Painting Contractor for 35 years and have done many tie walls. First you will need to pressure wash the ties. I use 1 cup of bleach and a 1/2 cup of TSP mixed in a 3-gallon pump sprayer. I coat the ties fairly heavy and let sit for about 10 minutes. Next use a medium tip with your pressure washer, I use a turbo tip I bought at Sherwin Williams, make sure not to let the pressure washer dig into your ties.

After about a week of drying time you can apply a coating on them. If you are planning on painting I use Sherwin Williams Duration, this is a great paint and primer min but will take 2 coats.

If you plan on using a Latex paint, I recommend first a good oil base primer, have the store tint the primer close to your finish color.

And actually, what I use is a driveway sealer, it will give it a nice black finish and 1 coat usually does the trick.

Before you paint it's a good idea to fill any large cracks and holes with a good 100% latex caulk, make sure it says "Paintable" on the label, silicone will not take paint and the paint will spread after a few minutes.