Its an interesting question and as has ben pointed out already - it really all depends on what you are wanting to achieve, if you can express this in more detail.
I have done a lot of practical work with active antennas as a hobbyist. Ultimately what I have been interested in has been real world comparisons of wanted signals versus noise/intermod in comparison with reference antennas. This has given me most information about the success or otherwise of particular designs.
But that is not very rigorous. To actually make meaningful measurements you would want to consider that there are two principal parts to an active antenna - the antenna itself, and the active bit - i.e. an amplifier. The antenna should ideally be matched to the amplifier although in return loss terms this is nowhere near as critical, nor at times, even important as with antennas used for transmitting. When transmitting you are usually interesting in maximum power transfer, for receiving this is often not as important as other considerations.
The antenna part is like any other antenna and can be analysed with a VNA. This can tell you about its complex impedance at given frequencies so you can ensure that your amplifier design is appropriate for it. You will have the problem that for many types of antennas the impedance may be way beyond what your test gear can sensibly cope with.
You can check amplifier design by making S11 measurements with a VNA for complex impedance, and gain by using S21. Again, a complicating factor is whether your VNA can cope with the range of complex impedance your amplifier is presenting.
My own experience is with active antennas for lower than a few hundred Mhz. I have never actually gone so far as to provide accurate conjugate match been the antenna and the amplifier - normally I am only concerned about the order of magnitude - i.e. if the antenna is a voltager probe type, is the amplifier a high input impedance. Some folk might need to do all of this in a less adhoc way and my hat is off to them - they have more knowledge than I do.
Another measurement is of noise. factor of the amplifier. Again, for me, it usually isn't too hard to ensure that the noise factor is a magnitude better than the noise presented by the antenna part so I don't see much need for measurement (can you hear the antenna noise when you connect it ? If yes, your amp has low enough noise factor). But if you are working at the limits, then sophisticated noise analysis and measurement may be required. If you were working at the limits I guess you wouldn't be asking the question so I don't think this is an area you will get much benefit from following up.
To me the key measurement for active antennas is their inter modulation performance. Poor IMD can make an active antenna less than useless. Measure this with a spectrum analyser and two RF signal sources of adjustable and known level. Or do this somewhat empirically with an SDR connected to the antenna and watching/listening for IMD products to appear/disappear as you vary the input to the active part by means of a stepped attenuator.
Cheers
Eloso