Most likely yes. If you were convicted of DWI, the convictions will be on your record forever. It just depends on what database your employer uses to do background checks. A DWI conviction stays on your record forever.
Criminal records checks often reveal charges from decades past. Two issues are presented which may result in your favor. 1. Not all systems were updated before computers became commonplace. In some cases, those records were lost or destroyed. 2. In many employment records checks, the employer only requests information re the last 5-7 years. If this is the case, your DUI/DWIs will not show up on the records check. Some states limit the access to older convictions. You should contact a lawyer to see if you can have your record sealed or expunged.
Criminal convictions stay on your record forever absent some type of pardon or clearing of the record. It is likely that these convictions will show up on a background check.