Denote the median of $\max_{i=1,\dots,n}|x_i|$ on the sphere by $M_n$. It is known that the ratio between $M_n$ and $\sqrt{\log n/n}$ is universally bounded and bounded away from zero. If you take $d=M_n$ then the quantity you are looking for is exactly $1/2$. It is also known that the $\infty$-norm ($\max_{i=1,\dots,n}|x_i|$) is ``well concentrated" on the sphere meaning in particular that for any $\epsilon>0$, if you take $d<(1-\epsilon)M_n$ the probability you're interested in tends to zero and if you take $d>(1+\epsilon)M_n$, it tends to 1. Quite precise estimates are known.
The way these things are estimated is by relating the uniform distribution on the sphere to the distribution of a standard Gaussian vector: If $g_1,\dots,g_n$ are independent standard Gaussian variables then the distribution of $$ \frac{(g_1,\dots,g_n)}{(\sum g_i^2)^{1/2}} $$ is equal to the uniform distribution on the sphere. So the quantity you're looking for is the probability that $\max_{i=1,\dots,n}|g_i|\le d (\sum g_i^2)^{1/2}$. Since $(\sum g_i^2)^{1/2}$ is very well concentrated near the constant $\sqrt n$, this is asymptotically the same as the probability that $\max_{i=1,\dots,n}|g_i|\le d \sqrt n$.
For general reference in a much more general setting you can look here: Milman, Schechtman, Asymptotic theory of finite-dimensional normed spaces. For a finer treatment of the special case of the $\infty$-norm, look here: G. Schechtman: http://www.wisdom.weizmann.ac.il/mathusers/gideon/papers/ranDv.pdfThe random version of Dvoretzky's theorem in $\ell_n^\infty$.
