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It would appear it does not but this is only sourced from me not having been able to any mention of them through my searches at: westeros, A Wiki of Ice and Fire, Google and reddit. I have also tweeted Elio and Linda, the co-authors of The World of Ice and Fire, and have yet had no response.

I have tweeted Elio and Linda, the co-authors of The World of Ice and Fire, and they have stated no nets.

@westerosorg Hello! I've found some people claiming that the 581st Lord Commander installed suicide nets on the Wall, though I haven't found any sources backing this up. Do you know if this is the case and if so was it the result of the 581st LC?
@westerosorg It seems I've found the answer to this (they don't) but I'd still appreciate some feedback if I'm wrong in my assessment. Thanks :)
You are quite right. That person claiming the 581st Lord Commander had them installed must have been very confused or was trolling.
_{Twitter, westerosorg}

• It's 700 ft high to successfully stop a suicide it would need more than one row of nets and each would be further out than the last
• Suicide = desertion and desertion is punishable by death (though the watch is undermanned now so there's some leniency e.g. Jon)
• The Wall is 300 miles long so that's a lot of nets to keep and maintain
• The watch is severely undermanned and can't even maintain the trees growing near the Wall or all their castles, the nets are secondary compared to this

• It's 700 ft high to successfully stop a suicide it would need more than one row of nets and each would be further out than the last
• Suicide = desertion and desertion is punishable by death (though the watch is undermanned now so there's some leniency e.g. Jon)
• The Wall is 300 miles long so that's a lot of nets to keep and maintain
• The watch is severely undermanned and can't even maintain the trees growing near the Wall or all their castles, the nets are secondary compared to this
• Don't want to protect the wildling climbers

There are also some reasons as to why it would have them:

• Someone falling off accidentally or regretting the jump but unable to stop it
• Someone being pushed off
• They're short on men so someone saved is better than lost

However, there are several reasons as to why the Wall won't have nets:

• It's 700 ft high to successfully stop a suicide it would need more than one row of nets and each would be further out than the last
• Suicide = desertion and desertion is punishable by death (though the watch is undermanned now so there's some leniency e.g. Jon)
• The Wall is 300 miles long so that's a lot of nets to keep and maintain
• The watch is severely undermanned and can't even maintain the trees growing near the Wall or all their castles, the nets are secondary compared to this