Let's consider what it would take to have unbreakable (effectively infinite) surface tension:

Either existing intermolecular forces would need to be dialed to infinity, or a new intermolecular force must come into action. In either case, it would make it energetically favourable for gaseous water to immediately condense into liquid whenever a gaseous molecule interacted with another water molecule. It would be an ice-ix scenario. All water would fall out of the atmosphere within hours, everything which uses lungs would find them filling with fluid. No water could be poured or create any droplet smaller than itself or otherwise separate from other water. However, that's not even the weirdest bit.

If this new or altered intermolecular force functionally increased the attractive forces between molecules of water, and only water, to infinity, all water would immediately collapse such that the individual atoms would undergo fusion, breaking the bonds of the molecules in a conflagration of nuclear fire.

But let's assume that it reaches just before the point at which the atomic bonds break. The water will likely take on the properties of a glass, becoming effectively solid, everywhere, just like ice-ix.

So let's be more generous and assume that the intermolecular forces are increased to be only strong enough to make it effectively impossible to break surface tension. We'd see a significantly higher viscosity, but what else?

Well, the intermolecular forces will probably still SIGNIFICANTLY decrease the solubility of pretty much everything, everywhere, all at once (but especially covalent gases, which do not dissociate).

This means that, in every living thing, at the same time, bubbles of oxygen and nitrogen will be coming out in the blood/hemolymph/cell membranes, not only making respiration functionally impossible (or at the very least far less efficient), but also embolizing every living thing with the precipitated gases. Everything alive dies, immediately.

If those two gases aren't enough, it will probably also significantly change the dissociation constants of pretty much every ionic compound, making them far less likely to dissociate in water, effectively causing large portions of the salt in the sea and other dissolved solids to precipitate in a cloud of powdered solids that would make the banded iron formations of the great oxygenation event look like a child's sandbox.

Depending on the interrelation of water's own dissociation and the intermolecular forces, which I can't recall at the moment, all acids and bases may suddenly neutralise in a similar event.

No matter what, I don't think anyone would be worrying about swimmers not being able to break the surface of the water.