Okay, so basically the problem lies with the hat that you took from Macgreggor's (or whatever his name was) body and gave to the sister in the bar for the debt collectors quest. What you have to do is go to the Crimson Caravan place (right next to the New Vegas medical place). Once inside, walk forward a bit and you should see on the right, a couple shelves with supplies on them. Next to the First Aid Boxes are two hats, kneel to see if you're detected or hidden, if you don't want to get caught, and take the "Old Cowboy Hat" and return to Freeside.
This is the most important part: MAKE SURE YOU'RE WEARING THE HAT AS YOU ENTER THE STRIP. Otherwise, it doesn't work. You should now be able to get in, but DON'T THROW AWAY THE HAT.
Hopefully now that we know what the issue is, Bethesda can create a patch so we don't have to wear those hats whenever we want to get into The Strip, but at least you can still progress in your game.
Hope this works for you too!
Edit: OTHER "OLD COWBOY HAT" LOCATIONS:
1. Mick and Ralph's in Freeside has three of them if you talk to Mick.
2. Powder Ganger Camp West inside the camper.
ALSO, make sure you don't leave the hat in The Strip as suggested somewhere in this thread. It works for a while, but then it stops working, and if you don't have another hat, you're pretty much screwed.
