A question regarding next-bit predictors
[SEE UPDATE BELOW] This is a very interesting question. Basically what it means is that if there exists a next-bit predictor, then there is a canonical distinguisher $D$ that outputs the $k$th bit and does nothing else. This is a good distinguisher since the $k$th bit equals 1 with probability that is polynomially bounded away from a random bit. I find this intuitively appealing and was first surprised. However, it does make sense.
As a sanity check, let's see what would happen if on the extreme:
$$
Pr_{X\sim D}[X_k=1 \mid X_1X_2\ldots X_{k-1}] = Pr_{X\sim D}[X_k=0 \mid X_1X_2\ldots X_{k-1}].
$$
This would mean that $X_k$ is actually independent of $X_1,\ldots,X_{k-1}$ and is a random bit. In such a case, it is clear that $A$ could only predict the next bit with probability $\frac12$. As a result, I think that this could be proven in the contrapositive.
I won't do all of this here; there is good value in you working it out. In order to get the direction, I propose that you look at the proof of equivalence of next-bit unpredictability and pseudo randomness. One place to see this is Section 7.3.2 of this chapter on Pseudorandomness by Salil Vadhan.
Let me know if it doesn't work out from here.
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I'm leaving the above for the didactic value, but I'm no longer sure that this is correct. The problem is the conditioning. In particular, once we condition on $X_1,\ldots,X_{k-1}$, then we are completely changing the probability space. (General recommendation - it's really easy to get conditioning wrong so stay away if you can. This was a recommendation given to me by my advisor. You can't always avoid it, but you need to be really careful.) In more detail, it could be that for \emph{many} given $X_1,\ldots,X_{k-1}$, the $k$th bit $X_k$ is polynomially biased away from $1/2$. The problem is that you don't know how to recognize the prefixes where this holds. Of course, this means that it can't happen with high probability (or a non-uniform distinguisher could hold some prefixes as advice). I think part of the confusion is what it even means to sample a full $X$ and then condition on $X_1,\ldots,X_{k-1}$. I think what you mean is that: for every $X_1,\ldots,X_{k-1}$, the probability of $X_k=1$ when the sample of $X$ has the given prefix of $X_1,\ldots,X_{k-1}$ is bounded away from $1/2$. I don't think this would be correct. If you mean something else, then it's worth thinking about.
I guess the important question is what you are trying to show and why, or possibly it's just understanding (which is good within itself).