You should look at the paper by Erdos and Odlyzko "On the density of odd integers of the form $(p − 1)/2^n$ and related questions" in Journal of number theory, vol. 11 (1979) pp 257-263. Among other things they prove the the set of odd divisors of $p-1$ (where $p$ runs over primes) has a positive density.
As far as the largest prime factor of $p-1$ there is a result of Fouvry "Theoreme de Brun-Titchmarsh: application au theoreme de Fermat", Invent. Math. v. 79 (1985) pp 383-407, in which he proves that the set of primes $p$ for which the largest prime factor of $p-1$ is $\ge p^{.6687}$ has positive relative density in the set of all primes.
