We had a hand rail up next to the door but it got pulled out which stripped the screws from the framing.  I assume that there are 2x somethings that frame the door correct?  What type of screws should I use to mount this handrail so that it doesn't pull out again?  I had thought to drill all the way through to the inside, use a flat metal bracket then bolts all the way through.  I'd put the smooth head on the inside and the nut on the outside.  Would this work?

Probably so, but be sure you seal the bolt holes so there is no chance of water migration from the outside. Use  good high grade bolts as most places sell cheap grade 3 bolts. I would suggest at least grade 5 but grade 8 might be overkill.

Thanks Racer-guy.

I mounted ours with bolts all the way through the side wall to a plate on the inside of the 5th wheel. In 15 years of use, it hasn't come loose.

Probably not a bad idea to use stainless steel nuts and bolts.