Renshuu’s sentence database is great for these kinds of questions. There are several hundred example sentences that use 多い, and even more that use たくさん.
As you can see in the examples, typical usage of 多い looks like 雲が多い.
My former Japanese teacher (native Japanese speaker) recommends using たくさん for objects and 多い for people. She says it generally feels more natural this way, although it isn’t a strict rule.
