The Enterprise uses Bussard-Collectors to harvest hydrogen from interstellar space.
While the Enterprise is science fiction, the Bussard-Collector - or more the Bussard ramjet1 - is a theoretical concept from a physicist named Bussard from 1960. In this concept, the spacecraft uses very strong EM fields to deflect and focus the hydrogen it's flying through into a (nuclear) reactor. Inside this reactor, the hydrogen is heated (accelerated) and jetted out backwards, which results in forward thrust.
However, Bussard's concept is more about how much hydrogen must be collected, how much it must be heated, and what initial speed is needed. It does not say anything about the field geometry, its strength or how to generate it.
It has to be noted that the fields must be incredible strong, because a not charged hydrogen molecule would interact with this fields only due to its (comparable weak) dipole momentum.
Matter consists of elementary particles, and for each of this particle types, an anti-type exists.
Positrons as anti-particle of the electrons are the most prominent ones and are for example released during a nuclear decay mechanism called inverse beta decay. Sodium-22 is an isotope which decays this way and is used PET, a medical diagnostics method. (Yes, anti-matter in your body!)
Neutrons and protons are not elementary, but consist of quarks. If you manage to put together three quarks, each being the counter-part of a quark in a neutron, you would get an anti-neutron. But this is really hard to achieve, due to physical constraints.
And once you have an anti-proton, it's also hard to combine it with a positron to an anti-hydrogen atom.
The PAMELA experiment states that the ratio of anti-protons to protons is 1:10000 in the van Allen belt. As to my knowledge, there are about 11g of protons in the belt, there should be 1.1mg of anti-protons. If you collect all of them, you get the energy a 1GW power plant delivers in three minutes - yet not so much.
Other sources state 10µg of anti-protons, and the answer from @Mark Adler states even less.
However... if your anti-matter actually consists of anti-protons or positrons (i.e. charged particles), it's (a little) easier to collect them. The fields must not be that strong, and it's easy to separate protons, anti-protons, electrons and positrons by their mass and charge. May be, you read about mass spectrometers to get an idea on how this might work.
Storing antimatter is another issue. If it's charged, you can use magnetic mirrors - in fact, the van Allen belt is a giant magnetic mirror.
As said, the amount of antimatter in the van Allen belt of the earth is quite small, you may have more luck with an other planet around an other star.
Stars also don't contain much anti-matter, but (as Mark Adler wrote), it may be possible (and is subject to current research), that there are larger amounts of pure anti-matter somewhere out there.
1) Today, a ramjet is a special type of aircraft engine without any rotating parts. Air is compressed by ram pressure at the inlet only, not by a fan. Fuel is added and burned, and the hot gas leaves through the exhaust nozzle, providing thrust. But the aircraft must already be moving, before this engine can be started. When it is, it allows extreme high speeds.