The standard deviation for the sample proportion is given by $$\sqrt{\frac{p(1-p)}{n}}$$, assuming a sufficiently large sample.

For the normal approximation to hold, both $$np$$ and $$n(1-p)$$ should be at least 10.

The standard deviation of the sampling distribution for sample proportions is independent of the sample size.

Increasing the sample size decreases the standard error of the sample proportion.