Very simple: don't burn the oil.
Different fats start to burn at different temperatures (called "smoke point"). This can be as low as ~100°C for unrefined vegetable oils. Butter is also low, at around ~150°C, which is below frying temperatures. Refined vegetable oils have very differentn smoke points depending on the plant they were made from, some starting as low as butter, and going up to around ~250°C (safflower oil, rice bran oil). Fats meant for deep frying (palm fat, clarified butter) are also around ~250°C.
So the two things you have to do is 1.Choose the right fat, and 2. don't let the pan get too hot.
From a nutritional point of view, deep-frying fats are not so good, so I'd recommend using a vegetable oil with a high smoke point. If you can find refined safflower, it's perfect, but if not, canola is also a good choice.
About the temperature, the easiest way is to measure it so you will know how hot the pan gets at any given moment. I use an infrared thermometer when frying, and make crepes at 200°C. If you don't have one handy, you could try heating a pan with oil on a middle setting for a long time (20-30 min), look if it smokes. If yes, let it cool down, and start again with a lower setting. If not, increase the setting and wait again. When you have found out the setting where your oil doesn't smoke, you can use it, or you can try one setting above (the first where the pan starts to smoke) because the room-temperature batter cools the oil. It is time-consuming, but you only have to do it one to find out the correct temperature.
You could use a nonstick pan, but the typical PTFE coated ones lose their nonstick layer at about 250°C, and a pan used for frying quickly heats much more than that. It is safe when used at lower temperatures, but you cannot fry crepes at 150°C. So unless you have a ceramic-coated nonstick pan (but not the type which disintegrates from hot starch in oil!) or you are controlling the temperature of the pan closely with a thermometer, it is better to use a noncoated pan.
If you want your crepes to taste like butter, don't fry them in butter. Instead, keep a cup with melted butter in a water bath near the pancakes. As soon as one is ready, take it out of the pan and use a glazing brush to brush it lightly with butter on both sides.