In the general case when $n=\dim X$ is arbitrary Kollár proves an analogue for minimal varieties of general type, although the bound is much worse than in the surface case. In the situation at hand it implies that $\vert ((n+3)!)K_X\vert$ is basepoint-free (actually one can write $(n+2)(n+2)!$ instead of $(n+3)!$, but this is shorter).
See: Kollár, János Effective base point freeness. Math. Ann. 296 (1993), no. 4, 595–605.
A closely related problem is Fujita's conjecture, which states that if $L$ is an ample divisor on $X$, then $|K_X+mL|$ is basepoint-free if $m\geq \dim X+1$ and very ample if $m\geq \dim X +2$.
There are many related results, but the conjecture is still open.
Angehrn and Siu prove that if $L$ is ample, then $|mL+K_X|$ is base point free if $m\geq 1/2(n^2+n+2)$\frac 12(n^2+n+2)$.
Angehrn, Urban; Siu, Yum Tong Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291–308. 32J25 (14C20 32L10)
Helmke also has some related results:
Helmke, Stefan . The base point free theorem and the Fujita conjecture. School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 215--248, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001.
Helmke, Stefan . On global generation of adjoint linear systems. Math. Ann. 313 (1999), no. 4, 635--652.
Helmke, Stefan . On Fujita's conjecture. Duke Math. J. 88 (1997), no. 2, 201--216.