On our old kitchen tap handles, the cold is on the left side and hot on the right. The new taps are the exact opposite, cold being on the right and hot on the left side. Is it okay to just connect them as they are as long as we, obviously, remember what side the cold and hot are on? Can it ruin anything like the cold water hose/pipes, etc.?
The biggest risk is someone being burned. In North America, cold should always be on the right side and hot on the left. If you are outside of this region you should check with local standards.
It sounds like you either bought a faucet from a different region, or maybe your original faucet was installed backwards.
With regards to the actual plumbing, it shouldn't make a difference which side you are hooked up to.
This is an easy fix. Your supply lines were simply installed to the wrong shutoff.
If you look under your sink there should be two shutoff valves. One hot, one cold. The supply lines (may be rubber or have a stainless sheath or plastic) go from these valves to your faucet.
Turn the shutoffs off. Open both sides of the faucet to relieve pressure. Unhook the lines at the shutoff and put them on the opposite shutoff. Turn the shutoffs back on. You should now have cold on the right and hot on the left.
This is such a simple repair there is no reason to "just live with it" and as @Steven states it could be a minor hazard for guests.