Generate sequences of numbers continually, ad infinitum, and determine if distribution rates are (?).
How would one begin to fill in that question mark? A PRNG that spits out three million 'zeros' and a single 'four' is as reliable as a PRNG that spits out an even distribution of random numbers.
The issue here is the entropy of such a sequence is extremely low, and while low entropy PRNGs do exist, and they are not considered good for cryptographic purposes. If algorithmic weaknesses led to long stable and highly compressible sequences, or rather, sequences with a very low entropy (as measured by say, Kolmogorov complexity) would be considered suspect.
Determining Kolmogorov complexity is itself a difficult problem, it's the problem of finding the shortest program (in a theoretical computer science sense) that produces a given output. And the complexity of that sequence is trivial, which is a huge problem for PRNGs. If every other seed had an extraordinarily high complexity with lots of hidden state, then maybe it would be considered, but I suspect not.
That is, if you accept a certain probability of falsely identifying a RNG as biased, I don’t see why this should be a fundamentally difficult problem, at the very least for the generator with four million 0s and a single 1: The probability that a good RNG spits out less than a thousand 1s in four million queries is negligible compared to the probability that it returns between 1.5 and 2.5 million 1s.
Of course, it will be harder to detect more sophisticatedly broken generators, but still, without requiring certainty, it should be possible to make some sensible claims?
For practical help, see http://www.stat.fsu.edu/pub/diehard/ (linked from http://en.wikipedia.org/wiki/Diehard_tests)
Any number or group of numbers that are consistently over/under represented should point to a bias as far as I can tell.