Based on their respective performances this year, the answer may actually be no. Make no mistake, Gardner is having a break-out year, and should have been given more All-Star consideration than he received. The problem is that Crawford has been better -- more than two times better, according to Fangraphs.
Brett gets on base more than Crawford, and that's about it. In particular, Crawford hits for more power than Gardner (and Alex Rodriguez and Mark Teixeira so far this year, but who's keeping track?) and plays much better defense, if you trust UZR (which I don't entirely, but that's the best stat we have). In total, were the Yankees to sign Crawford this offseason they'd be upgrading. In fact, the Bombers would be adding perhaps the 2nd-best player in baseball.
Always one to beat a dead horse, allow me once again to argue against signing Crawford. First, this smells like a contract year for Carl. Crawford is posting career highs in, well, everything. If he maintains his pace in the second half he'll post his best AVG, OBP, SLG, wOBA and WAR. While it is expected for a player Crawford's age to peak around now, the degree to which he is peaking seems unsustainable. Prior to this year Crawford's career best wOBA was .368. This year he's at .399, a material improvement. Buyer beware with his statistics this year. The team may not get them in the future.
The second argument against Crawford returns to relative cost. The minimum to sign him next year will be $10 million per season, and probably more like $15 million per season. The Yankees have Javier Vazquez ($11 million), Andy Pettitte ($11 million) and Nick Johnson ($5.5 million) coming off the books. Not all of that $27.5 million, however, will be available for new talent. One of Javy or Pettitte will stay with the team (mark my words). Javy would be cheaper than Pettitte, but for argument's sake assume that either will cost another $11 million. Then, Nick Swisher, Robinson Cano and Curtis Granderson all get raises next year, totaling approximately $9 million. That leaves only $7.5 million for new players. Cliff Lee will cost at least $20 million per season. Unless the Yankees are looking to add $30 million in payroll the team will not sign both Lee and Crawford. Expect the team to make a second run at Lee while letting Crawford go. He may be better than Brett Gardner, but Grit will only cost $400,000 next year.