I have seen questions where surface tension causes extra force to be applied on a solid to lift it from a liquid so it must exert a downward force. On the other hand I know that surface tension helps certain insects to float on water this implies that it exerts an upward force. Is this due to Wetting or Angle of contact? How does either affect it?
It may help to first consider a bead on a horizontal string under tension, for which you can have a better intuition. If we don't touch the bead, the gravitational pull will be balanced by the upward tension in the string. Thus, the string tension in this case pushes upward. If instead we pull no the bead upward, there will be a resistance exerted by the tension again, which will now be directed downward.
Surface tension acts exactly in the same manner, except that you now have a 2D surface instead of a 1D string. Surface tension resists an increase in area of the liquid surface, just like the string tension resists an increase in length of the string. As you can see, surface tension can exert an upward or downward force.
As Manuel Fortin points out, the surface tension force applied to an object floating in a liquid can be either attractive (the fluid wets the object's surface) or repulsive (the fluid repels the object's surface). If this force is uniform all around the floating object's "water line", then no net force is exerted on the floating object, and it floats motionlessly.
However, if the force is not uniform, surface tension can actually propel a little floating object like a wood chip. Try this experiment: in a basin of water, drop a wood chip that has a tiny gobbet of bar soap pressed into one end of it. Observe what happens!