I don't think there's any area on Houghton Street that is outright dangerous. However, there is a definite distinction between the side toward Heywood Street, which is much more characterized by single-family homes and the side toward Grafton Street, which is much more densely-populated.

I assume you'll be towards the Grafton Street end, since you're renting an apartment. That section tends to be lower income, which often means social problems, and there is public housing on Addison Street, right off of Houghton. This is a turnoff for some people, but, honestly, I've walked around the area at all times and never ran into any problems with anyone. The bigger issue I'd worry about is your building: make sure the apartment is a decent apartment, no pests, etc.

ETA: I hope you have a car, because if you are going to one of the universities, the commute would be pretty bad by Worcester public transit. It's free, but you would take a long time to get to Clark or WPI from Houghton Street if you're relying on bus. If you're not driving, you should consider somewhere closer to the school.