To the best of my knowledge, there is no known result that applies to this case immediately. I'll mention some articles which demonstrate techniques that could be used.

Riordan and Selby (2000) found the exponential part of the probability that a random graph $G_{n,p}$ has all degrees greater than a given value close to the mean.
Interesting, they showed that if $P_n(p)$ is this probability, then $P_n(p)^{1/n}\to C$ for $n\to\infty$ with fixed $p$, where $C\approx 0.61023$ is *independent of $p$*.

McKay, Wanless and Wormald (2002) found the precise probability for the same problem, but only when $p$ is close to $\frac12$.

Ordentlich, Parvaresh, and Roth (2012) found a very good approximation of the probability that a random $n\times n$ binary matrix has all row and column sums at least $\frac12 n$.

You will see that the calculations are quite laborious even if only an approximation is needed. From the second one you can also see that the asymptotically exact answer can be complicated too.

In all these cases, the effect of non-independence is very strong. For example, in the third case, although one might guess a probability in the order of $2^{-2n}$, the actual probability is close to $2^{-1.46n}$. The row sums are independent, and the column sums are also independent, but the column sums conditional on the row sums being large are very different from the unconditional column sums.