Learn French: Nous avons installé la barrière de peur que le bébé tombe. - We installed the gate for fear that the baby might fall.

Tombe actually is the subjunctive here! Because tomber is a regular -er verb, its present subjunctive form for the third-person singular (que le bébé tombe) is completely identical to its present indicative form. The conjunction de peur que always triggers the subjunctive to express an uncertain feared outcome, so grammatically, this is the subjunctive mood.

Shouldn't there be a ne before tombe after de peur que?

You are thinking of the ne explétif, an untranslatable ne often placed after expressions of fear like de peur que or de crainte que (e.g., de peur que le bébé ne tombe). It is considered elegant in formal written French, but it is completely optional. In everyday spoken French, it is almost always dropped, which is why it is missing here.

Could I use the infinitive here and say de peur de tomber?

Not in this sentence, because the subjects of the two verbs are different. You use de peur de followed by an infinitive only when the subject is the same for both actions (e.g., Il marche lentement de peur de tomber - He walks slowly for fear of falling). Here, "we" (nous) installed the gate, but "the baby" (le bébé) is the one who might fall, so you must use the conjunction de peur que followed by a new subject and a conjugated verb.