Of course, most of the shopping malls etc., do allow babies, even in LV! There are several nice shopping areas, of which the nicest is probably the Forum Shops at Caeser's Palace. It's kid friendly and has some interesting visual stuff the baby might like.
There is a local babysitting service called Nannies and Grannies that people have recommended; should be in the yellow pages. Most but not all the restaurants are baby friendly but of course babies are never welcome in the casino.....