(bypassing the questions of whether anarchy is a form of government--which it is not, of course, or whether such a thing as a populace would exist in an anarchist situation, or what exactly would be considered to be crime, etc...)
different groups of people would deal with problems differently, presumably.
some anarchists support the idea of a highly organized anarchist society, with federations and councils and federated councils.
other anarchists want a simpler, more accessible society, in which personal relationships are more significant than policies.
think of what it would be like if you and your friends could (and had to) deal with problems directly, rather than appealing to someone with more authority than you. consider, for example, that the reason that organized crime is as successful as it is, is because they buy off the people at the top. this corruption at the top is a big part of why anarchists believe in not having a top.
far from believing in the goodness of everyone (which we are sometimes accused of), we believe that anyone will be corrupted by having too much power, so we fight against the accumulation of power, rather than trying to make sure only good people have accumulated power.
edit: continuing the answer i guess...
bowshooter, i'm not sure, given your comment below, why you think i didn't understand what you were asking. but i will try to flesh out my response a bit.
a. i assume that most (not all) people commit serious crimes based on circumstances that would no longer exist in an anarchist society. the people who did act in ways that were seriously unacceptable to the group of people that they lived among would be addressed by that group of people. there are lots of ways to address messed up behavior, and one of the things that would be a mark of an appropriate culture is the capacity to address specific situations, including the characters of all the people involved, rather than having rules that have to be followed no matter what (three strikes and you're out would be an example of such a rule).
b. i don't believe that in an anarchist society it would be everyone in families against everyone else. or everyone alone against everyone else. i believe (i think most anarchists believe) that people want to and will live in groups of people with whom they pretty much agree and get along (this is one aspect of the "free association" part of the anarchist principles).
people wouldn't be "banding together against crime" because that implies crime as the cause of the banding together.
c. what does "adequate protection" mean? in this culture it means being forced to participate in allowing an armed force of people (including paying for them to exist)--who have no relationship to us, or care about us in any way--to occupy our neighborhoods, to commit violence against us that we are not allowed to combat, and to be the only recourse that we are supposed to take when problems reach a certain level.
ugh. i guess we need a question specifically about police?