Broadly speaking, there was no systematic advantage in experience of professional soldiers in the south.

If you look at any given West Point class, it's hard to find one where the majority of officers served in the Confederate armies. Many of the critical classes from the 1830s and 1840s are overwhelmingly Union.

The pre-war combat experience of Civil War officers was largely derived from the Mexican-American war. In this war, both northern and southern officers received their first state of combat and many who would join either side fought in both staff and line roles. Even the staff roles tended to be combat oriented, where officers conducted reconnaissance, filled in for line officers that became casualties, organized and lead assaults by detachments, set up offensive and defensive positions etc.

However in one sense, there is some truth to the claim. For the first large battle in the American Civil War, First Manassas, the two Confederate commanders did have an experience advantage. Confederate leader Gen. Joe Johnston had commanded a small regiment of skirmishers in the Mexican war. P.G.T. Beauregard had a very active staff role. Both men were wounded in the Mexican American war. Meanwhile the Union Commander at First Manassas, Gen. Irwin McDowell, also served in the Mexican war, and was at the battle of Buena Vista, but was on the staff for the most part and apparently considered himself a logistics specialist.

If leadership among Union generals was more deficient in the first part of the Civil War, it was not due to lack of combat experienced officers. Rather the blame should be placed on the U.S. political system for difficulty in finding the right men and putting them in the right role. For whatever reason, perhaps Jefferson Davis's personal military experience, the CSA performed better in it's choices for early war command.