In reality, .NET 3.5 is not a separate *version* - it is additions to the 2.* CLI. The main decision here is 2.* or 4.*. If this is a web application, you can configure IIS to decide which CLI to use.
So: if the "old versions" are 2.*, then installing .NET 3.5 **will** impact the existing applications, and the service packs **could** cause behaviour changes in some edge cases (pretty unlikely most of the time). More likely: the service packs that come with 3.5 will *fix* some existing BCL bugs.
A better option, though, might be to move your new app to 4.0; that **is** separate to the older versions, so you get an independent .NET engine. Perhaps just as important: you'll get all the current libraries and fixes.