Let me begin with what was formerly a comment above: the feeling among most experts is probably* that for each fixed $d$, the number of isogeny factors of $J_0(p)$ of dimension $d$ should be small compared to the dimension of $J_0(p)$, i.e. $o_d(p)$.

In what follows, I will cite some results which point in this direction. My overall response is not as logically coherent as I might have liked: perhaps a true expert will do better.

When $p$ is prime, it follows from a 1975 theorem of Ribet (reference below) that the $\mathbb{Q}$-rational endomorphism algebra of $J_0(p)$ is the same as the geometric (i.e., $\mathbb{C}$-rational) endomorphism algebra, and that this algebra is a product of formally real fields, each being the subfield of $\mathbb{Q}$ obtained by adjoining the Fourier coefficients of the various weight $2$ cuspforms of level $p$.

Thus the problem can be viewed as a special case of a popular one in pure algebraic geometry: for which genera $g$ do there exist complex algebraic curves of genus $g$ with, e.g., Jacobians isogenous to a product of elliptic curves? (Or, more generally, with endomorphism algebra containing at least $N$ semisimple factors?) If you count codimensions in the Siegel moduli space corresponding to (say) principally polarized abelian varieties with certain nontrivial endomorphism algebras and the Torelli locus (i.e., of Jacobians), then you find that (at least in many special cases) the sum of these codimensions adds up to more than $\frac{g(g+1)}{2}$, the dimension of the Siegel moduli space. Thus unless there is **excess intersection** between these two loci, for sufficiently large $g$ one expects Jacobians not to be highly decomposed. [This is a cue for Prof. JSE to weigh in on the matter.]

For instance, I think it is at least conjectured that for sufficiently large $g$, no $g$-dimensional Jacobian splits completely into a product of elliptic curves. This is known to be true over finite fields, but maybe not over $\mathbb{C}$.

Serre has worked on both the general geometric question and on this special case. In his 1997 JAMS paper, Serre showed that the maximal dimension of a simple isogeny factor of $J_0(N)$ approaches infinity with $N$. I think the result is quantitative, so if you look there you may get the most information currently known on the question you asked.

Ribet, Kenneth A. Endomorphisms of semi-stable abelian varieties over number fields. Ann. Math. (2) 101 (1975), 555--562.

Serre, Jean-Pierre
Répartition asymptotique des valeurs propres de l'opérateur de Hecke $T_p$. (French) [Asymptotic distribution of the eigenvalues of the Hecke operator $T_p$]
J. Amer. Math. Soc. 10 (1997), no. 1, 75--102.

*: I think it's true, and the small number of people I've spoken to about this explicitly (several years ago) thought it was true. We'll see what others have to say.