I agree with Santa. (What a lovely sentence that is.)
I know people whose relationships broke down because of one partner starting something online with someone else. But I think if it were not the internet it would have been the pub or the tennis court or something else that invited the wedge.
Going on my experience alone, I would say that the internet opens up a whole new world of fun, healthy friendships and romance. Of course one has to be careful when meeting new people but that is no different from meeting new people offline. Many people I know see the internet as a place to meet, play and learn but not as a replacement for in-person fun or business. Just an extension of it.
So to answer the question... I don't think the internet encourages secrecy or damages relationships. However, I do think it can be used as a tool for secrecy which can damage relationships but it's not the cause. The people are the cause. They use whatever tools available to them in any era.
My dad didn't have the internet in the 70s but he sure found ways to cheat on my mother, lie about it and have friends cover it.