The book by Barbour et al (whose title escapes me at the moment) discusses much more complicated versions of this; you might want to take a look at it.
For $k$ much smaller than $n$, there is a pretty simple observation.
Since the coefficients form a strongly unimodal (aka log concave) sequence, tails decay at least as fast (in fact a lot faster in the case of binomial coefficients) than geometric series, where the ratio of the second largest term in the tail to the second largest one is the $r$ for the geometric series, and the ratio is trivial to calculate.
This yields a surprisingly accurate estimate for the sum of the mass in the tail, and can be refined, using several terms. It is also easy to use when $k(n) = o(n)$ (combined with Stirling's formula).