"Play" or "playing" – which one is correct and why?

People often see him play basketball on the playground at the weekend.

Can I use playing instead?

We can use either the infinitive without to or the -ing form after the object of verbs such as hear, see, notice, watch. The infinitive without to often emphasises the whole action or event which someone hears or sees. The -ing form usually emphasises an action or event which is in progress or not yet completed.

• He saw her drive off with a young man in the passenger seat. (The speaker observed the whole event.)
• Maria heard him coming up the stairs towards her room, and felt scared. (The action was in progress, happening, but not completed.)

(from English Grammar Today)

Therefore, both forms are correct and have the same meaning. They might suggest different meanings when using them in past tense however.

Another thing:

After can or could with one of these verbs, we always use the -ing form, not the infinitive.