This is a very broad and not really answerable question, so I'll give you an answer for some very narrow definitions of "western music."

If you're asking when orchestras started incorporating a percussion section as standard in "western concert music" (AKA classical music), that really developed during the first few decades of the 19th century. Drums were frequently used in opera long before that to signal the entrance or soldiers, or to give an "oriental" sound to the music (more on that later). Otherwise, before the mid-1700s the kettledrum was the only percussion instrument that shows up regularly in the works of popular composers like Bach and Handel, and many well-known Baroque composers like Vivaldi didn't include percussion in any works.

In the mid-18th century, so-called "Janissary music" became quite popular, inspired by Turkish military bands. These bands featured many percussion instruments, from drums to cymbals to bells and triangles, and many classical composers started to incorporate these elements into their music, especially for an "oriental" effect. Mozart's Abuduction from the Seraglio uses these "Janissary instruments," namely the bass drum, cymbal, and triangle. Haydn included those instruments in his 100th symphony, also known as his "Military Symphony." He also made good use of the timpani in his later symphonies, and even started his 103rd off with a big drum roll. His 94th symphony is one of the earliest examples of a timpanist needing to change the pitch of the instrument mid-piece, a standard practice in orchestral music nowadays.

Beethoven is often credited with expanding the role of the percussion in an orchestra, bringing them from curiosities used to evoke a certain mood to an integral part of the ensemble's sound. All 9 of his symphonies include percussion, and the bass drum-cymbal-triangle combo is a big part of the epic sound in the finale of his 9th. By his death in 1827, timpani were standard in a symphonic orchestra, and the other percussion instruments were often used.

If you actually mean "western" as in "Country and Western Music," drums were really only standard in bands starting around the 1960s, with many notable exceptions. As this great blog post details, 1967 is usually the date that people bring up as the first time that drums were played at the Grand Ol Opry, but this is very unlikely. Bob Willis' band used drums extensively in the 40s, and Jimmie Rodgers even incorporated percussion in two recordings in 1929, "Any Old Time" and "Desert Blues". Still, it seems that drums were generally frowned upon by country purists, and the Grand Ol Opry ban (however well enforced) does give a sense of what many thought of them.

This started to change in the 1960s, with influences from rock and roll creeping into country. Johnny Cash's band officially added drummer WS Holland in 1960, going from the "Tennessee Two" to the "Tennessee Three," and Holland had played with Cash on recordings and performances for years before that. Drums were also a big part of the "Bakersfield Sound" of musicians like Buck Owens and Merle Haggard that burst onto the scene in the 60s. Nowadays, drums are as much a part of the country sound as guitars and fiddles.