What to do with under ripe figs?
Figs are climateric fruits that is, they will continue to ripen after being picked from the tree (because they will start releasing ehtylene). However, contrary to other climateric fruits (such as bananas) this only works if the fruit is picked in the later stage of its developement.
From: Regulatory mechanisms of ethylene biosynthesis in response to various stimuli during maturation and ripening in fig fruit (Ficus carica L.). - Owino et al., Plant Physiol Biochem. - 2006

The fig is a climacteric fruit and treatment with ethylene in the later part of stage II stimulates growth, accelerates the onset of the rise in respiration, induces ethylene production and promotes ripening. A typical respiration climacteric is not displayed by fruit harvested before the natural initiation of ripening on the tree. However fruit in which ripening had been initiated on the tree gave a normal climacteric rise and ripened fully after detachment from the tree. Thus in many aspects the fig behaves as a typical climacteric fruit, but as a result of its growth pattern, the ethylene production preceding ripening occur during a phase of active cell expansion.

What many people suggest, is to put some olive oil on the eye of the fig. I have never done it, but apparently you only need a very small amount around the eye to hasten ripening.
Here are two links to forums explaining this fig-oiling technique 1, 2
Apparently this will speed up a lot the ripening of the fruits (although some people seem to report they are less sweet).
This works because olive oil will increase the release of ethylene from figs.
From the same paper I cited above:

It has been known since the 3rd Century BC that a drop of olive oil applied to the ostiole (a process termed as Oleification) of the fig fruit stimulates growth and leads to uniform ripening of fruits. The application of several vegetable oils such as rape seed oil, sesame oil, soybean oil, camellia oil and linseed oil also had similar effects as the olive oil whereas animal oil stimulated fruit ripening but the effect was less than that of the vegetable oils. It was also observed that refined olive oil was more effective in accelerating maturation of the fig fruit as compared to unrefined olive oil or other vegetable oils.
[...]
A transient induction of ethylene was observed in fruit treated with olive oil, with ethylene production increasing a day after treatment and later declining