Lady with cage in the light by anne droogsma

A light cage is a somewhat older name for a woman of light morals, or prostitute. The origin of the word is uncertain; several explanations have been suggested. These do refer to shipping terminology, and it is even possible that the compound light cage as a whole is a sailor's term. This hypothesis is strengthened by the consideration that prostitution used to be particularly rampant in port cities; but it is not certain, and there are also other explanations for the word.

One explanation assumes light in a sense "easy".
Compare phrases like: "We will not forget that lightly,"
In that explanation, berth means "bed" (on a ship), as we still encounter in shipping terminology today.
In a lament on the death of Prince Maurice, Constantijn Huygens wrote:
Mauringh had crept to berth
And the endless sleep
Had beset his awake eye
And made him, lion to sheep.
(Meaning that Maurice retreated to his [dying] bed, and fell asleep.)
The light cage would then be someone easy to get into bed with.
Another explanation is given in DBNL. Now kooi would be a loan word, going back to the French cul (vulgar for "ass"), which was then corrupted into the Dutch word by a folk etymology. In this view, light is not an adjective, but a noun with verb tense: it would be related to "(op)lichten" in the sense of "lifting". The light cage then shakes her rear much, or lifts it: she "lights her cage".
Because light actually has not one meaning but several meanings, subsequent confusion may have arisen. At a later stage, the word part in this compound would still have been understood again as "not heavy", so roughly: "safe, easy of body".