Is there any halachic basis for requiring one to filter his internet? Does this fall under the category of "Ikah Darka Achrena Rasha Hu"? If a man is obligated to install a filter, does a woman have the same obligation?
The primary point of your question is the subject of contemporary discussion, and is hard to offer a "conclusion." At the asifa referred to in the comment on your question, it was accepted by many Rabbanim that it would be included in "ikah darka acharina," and it is difficult to find a good reason to differentiate it from that.
Regarding your final question, the Sefer HaChinuch mitzvah 387 says that לא תתורו אחרי לבבכם ואחרי עיניכם applies equally to women as to men. Iggros Moshe Even Haezer 1:69 says that women are not allowed to look at inappropriate pictures in films, and inappropriate internet images would be no different. R' Moshe (ibid) also applies this to seeing inappropriate behavior, and that would seem to extend to reading inappropriate material (since the logic is the thought process it instigates). So if men need a filter for the aformentioned reasons, women would also.