Steps

1. Choose a spot where you and your wife/girlfriend will be comfortable. Try to avoid very loud arenas or bars where people are loud and obnoxious. When you're not into sports, it can be annoying to be around people who are fanatic about a certain team. When you start watching sports with her, try to do it at home, with a few people she already feels comfortable with (not your buddies who always spill beer on the couch). You want her to feel relaxed, and associate sports with down time, not time to clean up after you and your friends, or time to watch the kids while you watch the game, or time to put up with incessant howling and cheers.
2. Choose a sport where your wife/girlfriend will have a pleasant view. You're not the only one who enjoys eye candy. Sports which display well muscled torsos or legs might be a good introductory choice. Don't pretend you don't know which sports have the most attractive athletes. If you can get her into a sport that you're not particularly crazy about, it might be a stepping stone to your favorite sport. If she's going to take an interest in something she normally wouldn't, you should be willing to as well.
   Good examples of sports which show off the male physique might include soccer, gymnastics, basketball, and swimming/diving. Avoid sports in which the object is to cause injury (boxing, mixed martial arts) or sports in which the players often get into fistfights, as these are the stereotypes that cause many people to think athletes aren't worth watching.
   Eventually, you can convince her to watch or go to certain games when her "favorite" athlete's going to be in it.^[1]
3. Choose a sport where technique and skill are emphasized over brute strength. American football may be fine for you, but most gals will likely get more excited about a well executed high bar release move in a gymnastics competition, or a triple lutz in ice skating. Gymnastics, Greek wrestling, ice skating (no hockey... sorry!), cycling, tennis, logging/lumberjack competitions, basketball, diving, swimming and the like might be good starting places. Choose a sport in which she can be easily impressed without being familiar with the game rules or scoring system.
4. Fill her in. With whatever sport you watch with her, explain the strategy or technique going on. Many people don't watch sports because they don't understand how the game works, so while you're cheering because your team just took a huge step towards winning, your significant other might still be confused, trying to figure out what just happened. It's no fun to feel out of the loop. If she's interested enough to ask questions, answer them all. Don't brush her off. Make an investment, here. Be patient and use a friendly tone of voice, even if she keeps forgetting which team is which. She's not trying to annoy you, she's trying to understand what's going on.
5. Play sports with her. Ask her to play a little soccer, football, or basketball with you. This is another opportunity to teach her about the game. Once she sees how difficult a sport can be, she might appreciate the athletes more. Don't get really competitive, and don't bring your friends. And don't be bossy. Make this a fun time for her. Crack some jokes. Tell her that you'll (insert chore here) whenever she scores a goal or makes a basket.
6. Choose a sport in which women can excel. Things like cycling and basketball have all female teams that your wife/girlfriend could root for and follow in the standings. This can be a way to introduce her to the sport and get her to watch with you.
7. Remind her that this is quality time together. Most of the time, when you watch sports, you're probably with your friends and buddies. Remind your girlfriend or wife that this is a great way to spend more time together, which is key to having a healthy relationship. And be fair--tell her that if she makes a genuine effort to enjoy watching sports, you'll make a sincere effort to appreciate something she likes to do. This brings us to the next step...
8. Barter. If she doesn't like sports because it's your excuse for not pulling your weight around the house (cleaning up, looking after the kids) or because she doesn't have any time to do the things she likes to do, then none of these steps will work. Help her wash the dishes or feed the kids before the game so you can both relax and watch sports together. Or, let her use your game time to do her own thing. At least she'll then see game time as a positive thing! Another idea is to "trade" quality time. If she watches this game with you, you'll watch a particular movie with her. Fair is fair!
9. Give her some team attire. Buy her flattering attire with your favorite team's logo. Buy what she is interested in. If she likes socks, get her some team booties. Even if she's not a material girl, the thoughtful gesture will most likely be appreciated, especially if it's a surprise. Encourage her to wear what you bought her and shower her with compliments: "My wife looks HOT in green, doesn't she?" or "You look incredibly sexy in that jersey."

Tips

• Find out if she's ever played sports before. If so, make that the first sport you watch together (presuming she liked playing that sport).
• While you're watching sports, you might be engrossed, but it really will help if you show her some affection. Hold her hand, stroke her hair, put your arm around her, give her a massage, anything to show her that you appreciate her spending this time with you.
• When you're first getting her into sports, be on your best behavior. Once she's into it, then you can display your usual enthusiasm. If she starts losing interest, however, follow the steps above again.
• Answer her questions! She'll have more appreciation for the game if she understands what's going on. Be patient. The rules of some sports are complicated for beginners.
• If possible, combine your first date to a sporting event with dinner at a restaurant she really likes.
• Find and relay stories that show athletes to be regular people, like funny stories of locker room pranks, the charity work they've done, or about their family life. Local newspapers are good sources for the personal lives of athletes on local teams, but make sure the stories and articles are positive.