Answer: 990

Small Hint:

If there are $r$ red and $b$ blue socks, it is equivalent to require probability $\frac{1}{2}$ of drawing one of each color

Big Hint:

Use $n=r+b$ and $d=r-b$ to turn the probability equation into a square condition

Solution:

Let $r$ and $b$ be the two color counts and $n=r+b.$ The probability of drawing different colors is also $\frac{1}{2},$ so $$\begin{aligned}\frac{rb}{\binom n2}&=\frac12,\\4rb&=n(n-1).\end{aligned}$$ Since $4rb=(r+b)^2-(r-b)^2,$ this becomes $(r-b)^2=n.$ Thus $n=k^2$ and, choosing red as the more numerous color, $$r=\frac{k^2+k}{2}.$$ The largest square at most $1991$ is $44^2=1936,$ giving $r=\frac{1936+44}{2}=990.$