"The president remains confident" that Snow will be confirmed, presidential spokesman Ari Fleischer told reporters aboard Air Force One.
White House officials said the administration learned about both incidents during the vetting process for Snow's nomination and believed neither matter had a bearing on Snow's Cabinet job.
"It's not relevant to his duties. We support him," Fleischer said.
Initial reaction from some of the senators who will vote on Snow's nomination was mixed. Some said the events occurred too long ago to have an impact, while others said they were withholding judgment until they learned more about the incidents. "We have to wait for the hearing," said Sen. Harry Reid, the Senate's No. 2 Democrat.
Snow was asked in a questionnaire from the Senate Finance Committee if he had ever been charged with a criminal offense.
"In 1982 I was arrested for driving under the influence of alcohol in West Valley City, Utah," Snow said. "I was never convicted of that charge and the prosecuting attorney voluntarily dismissed the charge before trial."
Snow said that in connection with the incident he paid a $334 fine "for making an unauthorized left turn with my automobile. I have never been charged with or convicted of any other offense."
In an addendum to the questionnaire, Snow disclosed that his ex-wife, Frederica Wheeler, sued him in Montgomery County, Md., in March 1988, alleging that he failed to pay child support and other costs associated with the care of his two sons.
Snow said he denied the charges, but the court found he failed to pay child support for his son Ian over a 19-month period and failed to pay Ian's transportation and allowance costs at college.
Snow told the committee that he and his ex-wife settled the dispute in January 1991 "to spare the family the difficulty of a trial."
His responses were released Tuesday.
Snow spokesman Dan Murphy said Snow would not have any further comment.
On Capitol Hill, Sen. Kent Conrad, D-N.D., said that "a DUI arrest 20 years ago is not material" to the nomination. Concerning the child support issue, Conrad said, "I don't know enough to pass judgment."
Conrad said his greater concern about Snow was his change in positions on controlling budget deficits. He said he would want to find out why "somebody concerned about deficits no longer is."
Sen. John Breaux, D-La., said that the drunken driving and child support issues had "nothing to do with his being secretary of Treasury. ... He's a good, solid pick. He's going to be confirmed, and I think he'll do a terrific job."
Mr. Bush picked Snow, chairman of the CSX Corp. railroad company, last month to replace his first Treasury secretary, Paul O'Neill, who was ousted in a Cabinet shake-up of the administration's economic team.
Snow had been expected to face tough questioning at his confirmation hearing Jan. 28 about Mr. Bush's new $674 billion economic stimulus program, which Democrats contend is weighted too heavily toward tax breaks for the wealthy and provides too little immediate support for the struggling economy.
On Dec. 9, the day the president announced his nomination, Snow sought to sidestep one possible controversy by announcing that he would resign from the Augusta National Golf Club, the host of the Masters golf tournament, which has come under fire for not admitting women. He also said would forego a lucrative severance package estimated to total up to $15 million that the CSX board could have awarded him.
Given last year's revelations about corporate accounting scandals, Snow was also expected to face questions about his management decisions as the head of CSX, the Richmond, Va.-based railroad he built into the largest freight line in the Eastern United States.