C
123 bytes, excluding #include lines.
int r,i;main(){srand(time(0));while(++i<9){r=rand()%62;if(r>35)r+=61;else if(r>9)r+=55;else r+=48;putchar(r);}putchar(10);}
Properly formatted:
#include <stdio.h>
#include <stdlib.h>
int r, i;
main() {
srand(time(0));
while (++i < 9) {
r = rand() % 62;
if (r > 35)
r += 61;
else if (r > 9)
r += 55;
else
r += 48;
putchar(r);
}
putchar(10);
}
Of course, you would want to use random() instead of rand() in a real application. I'm just shaving bytes.
Also, since 62 is not a power of 2, using mod slightly favors '0' and '1'.
EDIT:
Using the ternary operator (after muntoo's answer), 101 bytes:
int r,i;main(){srand(time(0));while(++i<9)putchar((r=rand()%62)<10?r+48:r<36?r+55:r+61);putchar(10);}