Many hotels have family rooms with three or four beds, large doubles with a sofa-bed or adjoining rooms with a connecting door. Practically all can provide cots, though sometimes a small extra charge applies. In some properties, smaller children (generally those under 12 years) stay free or are given a discount. If you want to find a nice family hotel, check this website.
Farmstays (Urlaub auf dem Bauernhof) are popular with families and offer a low-key, inexpensive experience. For best farmstay options, click here. Meanwhile, Heuhotels (hay hotels) offer the option of literally sleeping in a barn on a bed of hay: see this website (note that it's in German) for details. Camping is also huge; in summer the most popular sites book out far in advance.
Hostels, unfortunately, tend to have more of a party vibe and don't always welcome children. But there are still some family hostels present, look for them here.