What is the minimum sample size to get reliable results in haplotype and nucleotide estimations?
I'm still a little unclear on what the analysis looks like, but I'm going to assume for the sake of this answer that it's a single locus sequenced across multiple populations, with some number of individuals per population, and that the concern is that some of these populations are represented by only a few individuals. I'm also going to assume that the individuals are diploid, so you in fact have a little more data than it appears.

All the same, this paper indicates that n&lt;5 individuals in a population leads to quite noisy estimates of $\pi$ and $h$ (section "The effect of sample size on haplotype and nucleotide diversities"). This is of course specific to a certain ~500bp region that they collected sequences for. That doesn't mean that you can't compute these metrics, only that they may not be very reliable. 

For reference, the early Neanderthal genome papers compute some measures based on only the two copies present in their diploid genomes- of course, they are assisted by a large amount of modern human data for making comparisons. If you have supporting data from other populations, you may still be able to say something about the populations with few individuals. But you will likely have to use measures other than $\pi$ and $h$ to do so.

Some other tools that you might use to "borrow" information from other individuals outside those groups are ordination analyses such as PCA, or clustering tools like STRUCTURE. One way of thinking about how this works is that you can "embed" your populations with few individuals in the population structure of your overall dataset. This allows you to still learn from those populations even if the number of individuals is small. The Neanderthal papers implement versions of this, but with much more elaborate statistical frameworks, for a whole genome rather than just a single gene.

I'm not sure if DnaSP can carry out these analyses, but there are a lot of tools that do similar things, one tutorial here. Of course looking at the DnaSP manual is probably a good idea to see what you can do with your current tool.

Depending on your goals it may also be interesting to compute F-statistics across your populations, this is done by most population genetics tool suites. I used GenePop many years ago to do this, it appears to still work, though I am sure there are better alternatives now.

An important note is that most of these measures are purely descriptive, not inferential. Low sample size makes descriptive statistics noisy, but doesn't invalidate them. Inferential statistics are a bit more dangerous.