Recently there have been a few papers in this direction. The first one is AN ON-AVERAGE MAEDA-TYPE CONJECTURE IN THE LEVEL ASPECT by KIMBALL MARTIN. The main conjecture 1.1 of that article translated to this setting implies:
Conjecture: On average $J_0(p)$ has 2 irreducible componentssimple factors.
For $p >>0$ the decomposition of $J_0(p)$ always has at least two componentssimple factors since $J_0(p)^+$ and $J_0(p)^-$ are factors of it. So on average Kimball Martin one expects these to be the only two components. In particular for a fixed integer d the conjecture implies that on average $f(p,d) \to 0$ as $p \to \infty$.
Additionally Alex Cowan has done extensive computations on the decomposition of $J_0(p)$ into smaller factors for all primes $p < 1000000$. And his data only seem to support this conjecture. Indeed his data show the following:
Total number of simple dimension d factors for p < X
X d=1 2 3 4 5 6 Pi(x)
100000 1686 737 126 35 22 11 9592
200000 2751 1060 147 36 22 12 17984
300000 3717 1290 164 37 22 12 25997
400000 4592 1493 177 37 22 12 33860
500000 5427 1679 183 37 22 12 41538
600000 6226 1856 190 37 22 12 49098
700000 7020 1991 200 37 22 12 56543
800000 7728 2132 205 38 22 12 63951
900000 8456 2271 211 38 22 12 71274
1000000 9172 2412 218 38 22 12 78498
Where Pi(x)
is the prime counting function. In particular these data seem to be consistent with $f(p,d) \to 0$ on average.