it can affect their prices in two different ways:
1. everyone knows that whenever bitcoin is falling everything else falls harder. so ICO tokens are no exceptions. they can fall hard because of it and never recover.
2. which is the most important one in my opinion is the fact that when bitcoin is rising like last year there are a lot of newbies coming in with a lot of dreams of getting rich overnight while having 0 understanding of the market. and when price drops like these days the same newbies are running away as fast as possible.
these newbies are the main bulk of investors in ICOs and when there are more of them around ICOs can have better days than when there is less of them around and they are running away.
Well, that is a bit harsh on the newcomers. I would like to make it that not all newcomers behave the same way as expressed above, we don't have the actual numbers that support it, like research papers perhaps? Though there is truth in it, but I do not recommend blaming newcomers for it.
Let us also note that there are certain circumstances that when bitcoin price fluctuates, major investors (not necessarily newcomers) move their money from coin to coin thus affecting market price of certain coins or alts in general.