It has always been the case that anyone convicted of a crime can choose to forgo sentencing, if he pledges his life to the Night's Watch.
There are only a handful of people who still voluntarily take the black. Benjen Stark and Jon Snow are examples of this (it is somewhat of a tradition in the Stark family that younger siblings go to the Wall if they do not inherit the house, which is the main motivation for Benjen and Jon afaik).
Sam Tarly did not go to the wall voluntarily (Randyll Tarly forced him to go), but it's important to note that he did not commit any crime which caused him to take the black.
Around the time of Season 1, there are many allusions to how the Night's Watch is made up of nothing more than criminals. Even Jeor Mormont himself says so in the books:
A Game of Thrones - Tyrion III
"The Watch has no shortage of stableboys," Lord Mormont grumbled. "That seems to be all they send us these days. Stableboys and sneak thieves and rapers. Ser Alliser is an anointed knight, one of the few to take the black since I have been Lord Commander. He fought bravely at King's Landing."
Stable boys are not really criminals, but the underlying message is clear: the Night's Watch is mainly staffed by former criminals.
It stands to reason that the Wall was more or less considered as a form of community service, and when you stop taking in new recruits, you are essentially condemning more criminals to death. This may invoke the ire of the people, and may be hard to change.
The Wildling threat
During Jon Snow's time with the Wildlings, he is seen climbing the Wall. Wildlings still raided the lands south of the Wall, so the Night's Watch still had a purpose other than protecting against the White Walkers.
A Clash of Kings - Jon V
Restless, Jon squatted by the fire and poked at it with a stick. He could hear the Old Bear's voice inside the tent, punctuated by the raven's squawks and Qhorin Halfhand's quieter tones, but he could not make out the words. Alfyn Crowkiller dead, that's good. He was one of the bloodiest of the wildling raiders, taking his name from the black brothers he'd slain. So why does Qhorin sound so grave, after such a victory?
I'm struggling to find direct quotes about Wildling raiding. There are mentions by Old Nan, but she's an untrustworthy source as she's telling Bran scary stories.
However, you could argue that the usage of "Wildling raider" inherently implies that they perform their raids south of the Wall, since there is nothing to raid beyond the Wall.
This quote directly refers to wildlings going south of the wall:
A Clash of Kings - Bran II
Hother wanted ships. "There's wildlings stealing down from the north, more than I've ever seen before. They cross the Bay of Seals in little boats and wash up on our shores. The crows in Eastwatch are too few to stop them, and they go to ground quick as weasels. It's longships we need, aye, and strong men to sail them. The Greatjon took too many. Half our harvest is gone to seed for want of arms to swing the scythes."
Not only does this prove that the Wildlings raid south of the Wall, but it also highlights that the Night's Watch is considerably understaffed.