In fact, students of both German and Sanskrit say that the similarities are sometimes quite striking [I have this on good authority, but I have no examples].
I'm a German native speaker, but unfortunately I know nearly nothing about Sanskrit. Nevertheless, I deem it very unlikely that there is any significant similarity. Sanskrit has a full-fledged flectional system featuring about 700 conjugated verb forms, I think - German has exactly two conjugated tenses and only a handful of conjugated forms. Sanskrit has eight declensional cases, German three or four. -- This list could be continued, I guess, but perhaps your authority could set me straight there.
Really striking similarities exist (AFAIK) between Sanskrit and the classical European languages like ancient Greek or Latin. For example, Sanskrit has an optative mode and an aorist tense - like Greek, but unlike most other European languages (I don't know about Farsi). Then, I once found a Sanskrit conjugation table giving some forms of the verb "ni", "to lead" - the present has the forms "nayami", "nayasi", "nayati" (...) "nayamah", "nayatha", "nayanti". This reminds me stronlgy of Latin "eram", "eras", "erat", "eramus", "eratis", "erant". These forms aren't present forms, I know, but the principle is the same.
Finally, I think it is wrong to speak of a Sanskrit "influence" on European languages. It would be better to say that they are of common origin.
"Sanskrit" has a link to "Prakrit". When you go there, all it does is says "See Sanskrit." Nowhere is Prakrit defined. -- corvus13
Corvus, thanks for pointing that out. I made a stub; if you can add more, please do -- Be bold in updating pages.