The notion of enriched Reedy category is a combination of that of Reedy category and enriched category.
The main motivation for studying Reedy categories is that they induce Reedy model structures on functor categories.
The motivation for studying enriched Reedy categories is that they induced enriched Reedy model structures on enriched functor categories.
Enriched Reedy categories were introduced in
The defintion is def. 4.1 there.