Nothing much as per this paper by Ny Fed econs - Robert Battalio, Hamid Mehran, and Paul Schultz.
They compare two periods – first in 2008 post Lehman when there was a ban on short-selling in US. Second in 2011 when post US rating downgrade when there was no short-selling ban.
In September 2008, at a time of intense market stress, the United States and a number of other countries banned the short-selling of financial stocks. The bans were imposed because regulators feared that short-selling could drive the prices of those stocks to artificially low levels. Yet much remains to be understood about the effectiveness of such bans in stabilizing equity market prices. And reexamination of this issue is particularly important in light of the latest wave of bans in Europe, including the restrictions imposed by Spain and Italy in July.
Recent research on the 2008 bans allows us to assess the costs and benefits of short-selling restrictions. The preponderance of evidence suggests that the bans did little to slow the decline in the prices of financial stocks. In addition, the bans produced adverse side effects: Trading costs in equity and options markets increased, and stock and options prices uncoupled.
No blanket short-selling ban was in effect during August 2011, when Standard and Poor’s announced its downgrade of the U.S. bond rating. Our look at the sharp fall in U.S. equity prices following the announcement uncovers no evidence that the price decline was the result of short-selling. Indeed, stocks with large increases in short interest earned higher, not lower, returns during the first half of August 2011. Moreover, stocks that had triggered circuit-breaker restrictions and therefore could not be shorted on the day the downgrade was announced actually had lower returns than the stocks that were eligible for shorting.
The costs from banning short-selling in 2008 are estimated to be very large:
Together, the inflated costs of liquidity attributable to the shortsale ban in U.S. equity and options markets are estimated to exceed $1 billion. And, as noted earlier, this estimate ignores the lost gains from those trades that would have been made had bid-ask spreads been at or close to normal levels. The estimate also ignores the costs imposed on other markets. For example, convertible-bond arbitrageurs purchase more than 75 percent of primary issues of convertible debt (Choi et al. 2010) and hedge their purchases by shorting shares of stock. When the short-sale ban was imposed, the market for convertible bonds dried up (see Barr 
Overall, bans on short-selling do not work:
Taken as a whole, our research challenges the notion that banning short sales during market downturns limits share price declines. If anything, the bans seem to have the unwanted effects of raising trading costs, lowering market liquidity, and preventing short-sellers from rooting out cases of fraud and earnings manipulation. Thus, while short-sellers may bear bad news about companies’ prospects, they do not appear to be driving price declines in markets.
Interesting fin eco paper..