If I spin a ball attached to a string so that the string is pointing diagonally upwards, the centripetal force would be:
$$T \cos\theta,$$
where $\theta$ is the angle between the horizontal and the string, and $T$ is the tension. Then the vertical force would be: $$T \sin\theta + mg.$$
Given that the ball spins in a horizontal circle, how does it deal with an unbalanced vertical force and not fall downwards?
You should really start with the constraint $T \sin \theta = mg$. You know that the vertical forces have to be balanced! Then you can express $T$ as $T = \frac{mg}{\sin \theta}$.
And since the centripetal force is $F_c = T \sin \theta$, you can then say $F_c = mg \frac{\cos \theta}{\sin \theta} = mg \cot \theta$.
You can check that this makes sense! If $\theta = 90$, then the ball is at rest and there should be no centripetal force. Indeed $\cot 90 = 0$ so $F_c = 0$.
If $\theta = 0$, then the ball is completely horizontal. That's actually impossible, and you can see $\cot 90 \to \infty$, so the force would have to be infinite.
It does fall downward, a cycle with the angle $\theta$ maintained all the way around is not achievable.
If, however, it falls to $-\theta$ at an azimuth of $180^o$, (neglecting losses due to air resistance/friction and so on) it accelerates enough during that fall to return to $+\theta$ at an azimuth of $0^o$
This diagram should help:
The force of gravity is countered by the vertical component of the tension in the string, $T\sin\theta$; the centripetal force is provided by the horizontal components, $T\cos\theta$.
The string cannot be horizontal on a stable orbit with gravity.
You have a mistake. If you define the positive forces to up, rhen the gravity is negative.
$$T\sin\theta - mg=0$$ Or if you define the positive forces to down, $$-T\sin\theta + mg=0$$
Anyway, $$T\sin\theta = mg$$
Which means that the forces are in opposite directions. So, the force is not unbalanced force in the vertical direction.
