As others have noted, cat fountains are popular for this very reason. I don't doubt the typical feline fountain-lovin' behavior at all, but as an anecdotal aside: my kids leave the water running from the faucet from time to time, yet I've never noticed my "weird" cat taking any particular interest in drinking from it.
Cats Stop Purring at the Sound of Running Water - Pet Education
My kitty Nicky is not an older cat in fact he is 10 months old. Almost from the moment we got him we noticed he always played with his water, after being fed up with constantly wiping up his splashed water we began leaving his dish in the sink. He obsessively plays with it, turn on the sink he’ll come running from where ever he is. Leave the toilet seat up and he is head first up to his shoulders in the water. Turn on the tub or shower again he is right there. We have tried everything suggested and nothing has changed. All though all the advice is great, we finally have just concluded that Nicky is who he is, sweet, loving, funny and obsessed with
Cats often stop purring when they hear running water.
Many cats, including one of mine, refuse to drink out of a bowl. They prefer running water. Some drink from a slowly running faucet, but you don't want to leave a faucet running all day. Luckily, there are 'fountain bowls' available which can provide your cat with running water 24 hours a day.
Ghost the Cat and Running Water - YouTube
If you find that your cat runs to the sink everytime she hears you turn on the water faucet don't be surprised. Most cats prefer running water to water that is standing still, such as that found in their water bowl. Another theory (mine) is that water running from the faucet just tastes better than the water in your cat's bowl. After all, would you like to drink water from a cup or glass that has been sitting around all day? In addition, water coming from the faucet is usually colder than water that has been sitting in your cat's water bowl. One theory as to why cats prefer running water is that it may be due to an evolutionary process. Before humans kept cats as pets and provided water bowls full of fresh water, cats had to find their own sources of fresh drinking water. Running water is generally much healthier and less poluted than stagnant, still water, and so the cats that preferred to drink running water may have survived longer and more of their genes may have gotten into the kitty genepool. Getting cats to drink their full amount of water is essential for good kidney and urinary tract health. Water bowls are often not appealing enough for cats to drink their full amount, and running water can provide this encouragement. A small aquarium pump (2-3 watts) and a big ceramic, glass or steel bowl is all you need to set up a fountain. The commercial pet fountains work fine, but many cats are allergic to plastic and can develop a chin rash from contacting the plastic. There is also much more to clean; a simple pump in a bowl may not be as decorative, but it is much easier to clean than a more elaborate setup. Clean the pump with a toothbrush frequently. We stock these small pumps at Frederick Cat Vet.