Give them space. It is important to keep in mind that both of these crops are perennial and they both require a lot of space. Some artichoke varieties, like "Grand Buerre" can grow up to 9 feet tall. And, since they are thistles, you may not want them in a spot where you'll brush past them frequently. Asparagus are planted as root crowns and may take a year or two to become established. You may need a 10 to 20 foot row of space for an adequate spring crop of asparagus. Make sure mature plants won't eventually shade out annual garden plantings.
Give them the proper conditions. These plants have different soil needs and taking that into account will help ensure success. Asparagus are not picky about soil type but do not like to sit in water and are prone to root rot. Artichokes, on the other hand, prefer deep, rich soil that is well drained but always moist. According to Cornell University, artichokes require copious amounts of water to produce good buds, so you'll need to plant them where watering frequently won't be a problem and won't interfere with plants requiring dryer conditions.
Provide winter protection. Asparagus will do well in the winter, even in cold conditions, if allowed to fern out before the first freeze. Artichokes are more sensitive and will need protection in order to successfully overwinter. Cut them back to 12 inches and cover them with leaves or lawn mulch. Cover this with a box or bucket and add more yard debris on top.