I would definitely advise gating it off. When Hot Rod was a baby he had the run of the downstairs, at first, and I went into the other room and all of a sudden I heard a thud and a yelp. The only thing that I cant think of is that he fell trying to go up the stairs.

He was about 18 weeks old at the time and under 2 lbs. He is now 2 1/2 and is really just getting confidence in going up and down stairs and mind you, I follow him up with open arms and I walk down in front of him ready to catch him.
I have hardwood stair without any type of covering on them. We now have 2 baby gates tied up into a "C" shape at the bottom of the stairs to keep both of them off the stairs. Maggie has never been afraid of stairs and just bounds up and down! She is going to be 2.
BTW, congrats on you new baby!