I highly recommend this blog post from EFF on the very real problem of online harassment and the tremendous difficulty in crafting solutions that address it. Lengthy, and recognizing both the terrible impact online harassment can have and the dangers of overbroad solutions.
To sumarize, the danger of new laws, or in application of existing laws to online situations, is that we have poor enforcement both in terms of underenforcement (dismissing dangerous behavior by telling people to just ignore it) AND over enforcement against specific targeted populations (e.g., Muslims). Clearly better training and education in law enforcement needs to be part of the solution, and this should be emphaiszed rather than rushing to pass new laws which will likewise be over enforced and under enforced. Additionally, the concern that overbroad laws can themselves be used to bully and censor is real, with a demonstrated history unde existing law. This does not mean we should have no laws targeting specific types of harassing behavior, but that we must not view these as either easy to implement or sufficient in themselves.