I wouldnt...hes too young...growth plates still forming...you dont want him to fall and possibly break a leg or something...my puppy is 4 months old and i only carry him up and down the stairs...not worth the risk imo.
Linus is almost 5 months and only decided to do the stairs about a month ago. He does this rarely though and only when I come home from work and go up to change. He certainly doesn't run up and down them continuously or more than once a day... so we let him do it.
I wouldn't either. When they are young, you don't want to over stress their joints. Could cause hip displasia later. So I was told, years ago, when I ha a rottie puppy. Always carried him up and down.
We have always let our Bully babies go up and down stairs on their own. In our last house my office area was upstairs and the 3 different pups always went up there with me. There was no harm done; I didn't know better!
Molly has always loved stairs.. or to be exact, the things she finds upstairs (all the kids rooms are upstairs). We just make sure she comes down slow, or we carry her down.