It's a pity that the production of antimatter is so inconsequential.....something like 9 nanograms (9x10^-9 grams) a year. Given the way they produce it now (in particle accelerators), it's far too expensive to expand production of antimatter to usable quantities.
With the amount ot energy that's produced in matter/antimatter reactions, a rocket powered by this method would be able to attain quite reasonable velocities. Depending on how much reactants you mixed in the firing chambers of the rocket, the specific impulse of the rocket would achieve far more than unity (I think it's on the order of 1000:1 or more....for memory). With a reaction of 1:1 antimatter/matter (given it's a 100% matter to energy conversion), the thrust of such an engine would be spectacular. One gram of matter converted into energy generates the same explosive force as a large thermonuclear weapon, so you can imagine the propulsive thrust you could obtain.
It would be entirely conceivable to obtain velocities on the order of .75-.9 c using this type of engine. At those velocities, time dialational effects are minimal, but you would have to control the burn of the fuel to regulate the acceleration of the ship, especially when you had human occupants. For probes, it wouldn't matter so much, but you would still have to control the acceleration such that you didn't crush your probe by exceeding its design limits.
However, if we could create enough antimatter to make it a viable fuel source and designed an engine and containment system for the antimatter, it would be possible to reach all the stellar systems within 20 light years within a human lifetime (for probes on return journeys). Probes on one way journeys to systems they were to study, and that range would extend out to 30-40 light years. The communication times being the deciding factor in both cases. Unless, of course, a means to communicate faster than light was found. In which case 60 light years wouldn't be out of the question.
Other deciding factors to probe endurance, such as fuel load, provisions and life support (for manned vessels) and so forth, would be part of the design process and accounted for when determining thrust/acceleration and range of ship.