Should I enclose my 3D Printer? I have a home built RepRap with all sides open..

Would there be any advantage to enclosing the print area in acrylic?
It is hard to tell whether you personally should enclose your printer. However, you asked for the advantages and I will name some of them on which one can base a decision.

A 3D printer enclosure


helps to keep the temperature of the whole print at controlled levels, if you use a heating element, thermocouple and pid regulator. This is one of the most direct uses of the enclosure, which can be achieved by almost no other means. One could sloppily say it does for the whole print what the heatbed does for the initial layers. Controlling the temperature can be beneficial for layer adhesion and can help against delamination problems. This can go as far as fixing cracks and complete delamination (Thanks to @J. Roibal for bringing these cracks to my attention in the comments)
keeps dangerous fumes controlled. Here you can find a scientific study about it, published in Atmospheric Environment 79, titled 'ultrafine particle emission from desktop 3D printers, on exactly that topic. You can embed a filter with a fan in your housing to filter the air from all dangerous fumes that are created when melting certain plastic types. It could just circulate the air inside the chamber or get the filtered air out of the housing. This is another use which cannot be achieved otherwise (afaik).
can keep humidity away from your printer. This is helpful for filaments that attract water (and don't print well under that circumstance). This should be realized separately for stored filament, too, adding some silica gel to regulate humidity. (Thanks to @Obmerk Kronen in the comments)
minimizes losses of your heatbed. This happens in at least two ways, - the heated bed will also heat the surroundings, that is the inside of the enclosure. By raising its temperature, the temperature difference and hence heat loss is minimized. Also wind, introducing high fluctuations in the transfered (i.e. lost) heat is minimized. In that sense, it also
shuts out any wind for print temperature stability. Also dust and particles that could be blown on the print will be shut out (thanks to the addition of dust/particles: @Obmerk Kronen). This is a benefit that comes without having a heated chamber or filter.
helps to keep the printer clean in between use. Your axes will thank you being free from dust.
reduces smell and noise. If you use the printer in you living area, that alone can be a great benefit.
makes sure that your printer is safe during storage, nothing will fall on it.
can look pretty nice and add to the style of your printer, even if selfmade ;-) 


There are obviously also downsides, as: connected work/money to make it, increased space used for the printer, and, if not well made for that purpose (which it should be), increased difficulty in repairs and maintenance of the printer itself (i.e. to get the printer out of the enclosure).