Some plants such as cacti barely need any additional water to survive since they are extremely efficient at retaining the water they receive through their root system. On the other hand, it is near impossible to over water Bonsai provided they have the proper soil for good drainage.
Automatic greenhouse watering systems can be just like soaker hoses lying in or on the soil that slowly leech water into the ground. Alternatively, they can be overhead sprinklers that either spray or mist water onto the plants. Again, the type you want will hinge on the plants you intend to grow.
An intermediate solution is to simply have a water tap or long hosepipe from the house that lets you water by hand a lot more easily than using a watering can. In this scenario, an internal spigot or water tap is preferable so that you can keep the door to the greenhouse fully closed while watering.
Running a water pipe to the greenhouse is no more challenging than running normal lawn-sprinkler type PVC piping a few dozen yards. Unless you have a very big greenhouse, standard pipe will supply plenty of water for your needs. The pipes, couplers and controls required are reasonably priced with the only major expense being the digging of the trench. Of course, if you dig it yourself, the only cost will be on your back and not your pocket.
Take into account when planning any form of overhead watering system that mist or spray water, that some of the moisture is likely to fall in other areas besides the plants. Benches, floors and other home garden greenhouse components should be able to withstand wet conditions without causing problems.
Excess moisture will inevitably end up on the walls and roof, no matter what kind of watering system you use, especially if your plants are anything other than cacti. Water evaporates from the soil surface and plant leaves and humid air can get in when you open the door. Consequently, you should also plan for some type of ventilation.
Ventilation can be a simple fan, plus a narrow gap around (at least part of) the roof. Cleaning mildew and other consequences of moisture in the air can be done using standard products, but it will be done less often if you plan your greenhouse watering system not to accidentally spray anywhere other than the plants.
Allow around twenty percent of your greenhouse construction budget for your plumbing system. Even after the initial build, you will have to carry out maintenance from time to time. Pipes and hoses crack, especially in extreme temperature change conditions and valves eventually go bad.