A.R.
I think it all depends on what you want to do.
If you've never been to Las Vegas and are going for the 'Las Vegas' experience, don't stay downtown. Sure, they have the light show at night which is very cool, but downtown is way less 'touristy' than the strip (hence, why it is cheaper). I'm not sure there is a whole lot to do downtown with an almost 2 year old, because you certainly won't be taking him/her to the casinos.
If you want the glitzy, Las Vegas experience then stay on the strip and stay at a theme hotel. Most rooms in Las Vegas are 'relatively' cheap, so pick one ON the strip in the middle of the action and avoid the cheapies on the fringe. If you're looking for things to keep your little one entertained, think about Excalibur (lots of kid-stuff), Treasure Island (they do the pirate show, I think), and Mirage (they have the volcano). I've stayed at the Luxor and Bellagio (last time I was there, it was a 'no-kids' hotel).
Also, consider how much walking you want to do - if you're going to want to do a ton of sightseeing with an almost 2 year old, I would recommend picking a hotel in the middle of the strip. As much as I like the Luxor, it's on the far end of the strip and that's a lot of walking. Avoid cabs if at all possible; while they are soooooo much safer than Chicago cabdrivers could ever hope to be, the meter runs and runs and runs while you sit in traffic (which you WILL sit in traffic). On the upside, valet parking is FREE at every casino (just be nice and tip the dude).
I absolutely love going to Las Vegas - and I don't gamble much at all, maybe $200 on a weeklong trip. I go there for the food and wine (amazing!), the shows, the shopping, and the sightseeing. In other words, there will be plenty of activities other than gambling to keep your family busy and having fun!