Both are fine. Need, like dare, is sometimes called a "semi-modal", in that it can be used like modals (eg can, may) but can also be used as a normal verb (eglike want).

So The balance need be swayed only slightly is parallel to The balance can be swayed only slightly; whereas The balance needs to be swayed only slightly is parallel to The balance requires to be swayed only slightly.

I believe that the modal use is becoming less common, and I suspect it is altogether less common in US English than in UK, but I'm not sure about that.

Both are fine. Need, like dare, is sometimes called a "semi-modal", in that it can be used like modals (eg can, may) but can also be used as a normal verb (eg like want).

So The balance need be swayed only slightly is parallel to The balance can be swayed only slightly (the meaning is different - this is just showing the syntax); whereas The balance needs to be swayed only slightly is parallel to The balance requires to be swayed only slightly.

I believe that the modal use is becoming less common, and I suspect it is altogether less common in US English than in UK, but I'm not sure about that.