At the present time, the best (general) result is due to James Maynard, who has proven that any admissible 3-tuple takes at most 7 prime factors (see this paper). In particular, (using a certain admissible 3-tuple related to what Robert Israel wrote) we can get that $n(n^2-1)$ has at most 11 prime factors infinitely often.

I don't know if this number can be improved for the specific case you are interested in. However, the general consensus is that Dickson's conjecture is true, so both $p-1$ and $p+1$ should consist of exactly 3 prime factors, for infinitely many $p$. But this conjecture is currently out of reach of any known techniques, and appears to be harder than the twin prime conjecture, since we must control admissible 3-tuples, but the twin prime problem deals with only admissible 2-tuples.

At the present time, the best (general) result is due to James Maynard, who has proven that any admissible 3-tuple takes at most 7 prime factors infinitely often (see this paper). In particular, (using a certain admissible 3-tuple related to what Robert Israel wrote) we can get that $n(n^2-1)$ has at most 11 prime factors infinitely often.

I don't know if this number can be improved for the specific case you are interested in. However, the general consensus is that Dickson's conjecture is true, so both $p-1$ and $p+1$ should consist of exactly 3 prime factors, for infinitely many $p$. But this conjecture is currently out of reach of any known techniques, and appears to be harder than the twin prime conjecture, since we must control admissible 3-tuples, but the twin prime problem deals with only admissible 2-tuples.