I'm not sure what kind of information you're looking for, but your sets of primes are subject to the Hardy-Littlewood k-tuples conjecture.

Let $q:=k!$. The set of integers $n$ for which each of $\frac{n+1}{2},\dots,\frac{n+k-1}{k}$ is an integer is a union of residue classes modulo $q$. Let $a$ be such a residue class, and consider the $k$-tuple of linear forms $qn+a, \frac{q}{2}n+\frac{a+1}{2},\dots,\frac{q}{k}n+\frac{a+k-1}{k}$, and note that each linear form has integral coefficients by our choice of $q$ and $a$. To see whether this "should" represent infinitely many primes, one should check whether the product of the linear forms has a fixed prime divisor; I haven't done this, but I expect it doesn't, and certainly, there should be an $a$ for which it doesn't. Assuming that there is no fixed prime divisor, this tuple of forms is admissible, and we expect that there should be infinitely many $n$ for which each form is prime. Moreover, there should be a constant times $x/\log^k x$ such $n$ up to $x$. Using the Selberg sieve, one can prove an unconditional upper bound that is $2^k k!$ times as large. This is in Iwaniec and Kowalski, for example.

It's tempting to use the recent results of Zhang, Maynard, and Polymath to say something here, but we have to be careful. For example, we know that once $k$ is at least 51, there will be infinitely many $n$ for which at least two of the forms are prime. But this does not say that there are infinitely many primes $p$ for which one of $\frac{p+1}{2},\dots, \frac{p+k-1}{k}$, because we cannot guarantee that one of the two forms that simultaneously represent primes is the first form.

**ADDED:** Instead of looking mod $k!$, one can of course also look modulo $\mathrm{lcm}(1,2,\dots,k)=:L$, and restrict to $n$ that are $1 \pmod{L}$. This gives us some information about the size of the smallest element of the set. In particular, if we just look for the first prime $p \equiv 1 \pmod{L}$, this will give us a lower bound for the first prime in $H_k$. By Linnik's theorem, we know that the first such prime is $\leq L^c$ for some $c>0$; we know, due to Xylouris, that $c=5$ works. Of course, this is a coarser question than identifying the smallest element of $H_k$, but, given that I doubt we can prove that $H_k$ is non-empty for large $k$, I doubt we can do any better.