。
He stood frozen, gripped by the fear of falling.
Literal
Fall-will-surely-not? [quotative-という] [attributive-の] fear [by-で] he [topic-は] stood-frozen.
落ちはしないか is an emphatic rhetorical question: 落ちはしない ('will not fall' — emphatic negative with the は inserted after the masu-stem) + か (question). Combined with という, it functions as an embedded thought — 'the [thought/fear] that perhaps he might fall.' 立ちすくむ is a compound verb 立つ + すくむ ('freeze up / shrink back').