score:10
First of all one must know that free Sundays have recently been suppressed from April to September. The Louvre website says:
Admission on Sunday
From October to March: access to the permanent collections is free for all visitors on the first Sunday of each month. From April to September: no free admission on the first Sunday of each month.
The Louvre is always crowded, and you'll always have to queue at some point. The first Sunday of each month is no exception. Being a free admission day, it is often a day with a high visitor turnout. Having said this, there are strategies you can adopt in order to beat the crowds.
The first thing I would do is to pick a Sunday in a month in which there are no school holidays, neither in France nor in neighbouring European countries. If November 1st were to be a Sunday I'd never go to the Louvre since I'd expect it to be packed with tourists from all countries celebrating All Saint's Day.
Of course one can't always choose when the first Sunday of the month will be. The next invaluable piece of advice I can give is to pick an alternative entrance. Everyone queues at the Pyramid, because that, and the Mona Lisa, are what people of when they picture the Louvre. However, there are three other entries available, of which one, Passage Richelieu, is only for visitors who already have a ticket.
The other two entries are:
entrΓ©e du Carousel i.e. the entrance of the Carousel shopping mall at 99 rue de Rivoli. From there you can access the Louvre museum after going through a bag and security check.
Porte des Lions, located south-west of the Pyramid, close to where the Tuileries gardens begin.
See below for a detailed map (courtesy of Peccadille):
Louvre Map, Peccadille, Fair Use
The Porte des Lions is fairly off-centred from the main Louvre attractions, and will require you to walk a bit once inside. Personally, beating the crowd is worth the walk.
Finally, if you don't want to queue up at the Louvre, I somewhat agree with Peccadille who suggests to pick another museum. Since most paying-access museums offer free first-Sundays, why not try out one of the other, less known, such museums? Also, the musΓ©es de la ville de Paris are always free, and are likely to have less people visiting them on Sundays.