Muslim traders, both by land and sea, are recorded in Korean history from the late 7th century through to the 12th century. While there won't have been Muslim traders much before the earliest historical records of them, they probably continued to travel to Korea long after the 12th century - maritime trade with Muslim merchants is thought to have persisted into the 15th century. Some traders lived in Korea for extended periods, and some may have settled there permanently. Some Korean families claim descent from Muslim traders.

We can note the first verse of the famous Goryeo song "Ssanghwajeom" (the title is usually translated as "The Dumpling Shop" or "The Turkish Bakery"):

I go to the Huihui shop, buy a dumpling.

An old Hui holds me by the hand

If the story spreads out of the shop

where "Hui" means "Muslim". The song is from the reign of King Chungnyeol (end of 13th century). ("Holds me by the hand" is likely to be a later (Joseon) bowdlerisation.)

However, a decree of the 9th year of Sejong the Great (1427):

Since the hoehoe has worn different costume (with clothing and headgear), people watch them as not belonging to our people and avoid marrying them. Having become this kingdom's subject, our way of clothing must be followed in order to remove difference of the hoehoe. This would naturally lead to intermarriage. Furthermore, the hoehoe should be forbidden to do their way of 'rites' during Great Assembly of the Court.

appears to have led to the assimilation of Korean Muslims into the wider Korean society, and Islam as a Korean religion disappears from historical view.

In modern times, Islam has made three returns to Korea:

1. Conversion of Koreans living in Manchuria in the late 19th and 20th centuries; cities in Manchuria had substantial Muslim communities (e.g., Mukden/Shenyang had 5 mosques). Many Koreans living in Manchuria returned to Korea following the end of Japanese colonial rule (i.e., after WW2), including Korean Muslims.

2. During the Korean War, Turkish troops became aware of Korean Muslims, which resulted in shared worship at the time, and post-war support for the development of Korean Islam. This included the training of Korean clerics and the building of mosques (the first in Seoul in 1976, and others, mostly in Seoul, following).

3. Conversion of young Korean men working abroad in the Middle East.

To these, we could add Russian Muslims fleeing the Bolshevik takeover, some of whom moved to Korea.

I haven't seen anything on pre-modern Judaism in Korea.

References:

Little has been published on Islam in Korea. James Huntley Grayson, Korea - A Religious History, RoutledgeCurzon, 2002, has a very short chapter on the topic (1 page on Medieval Islam, and 1 page on modern Islam (19th century onwards).

More detail and discussion of modern Islam in Korea is available in Baker, Don, 2006, "Islam Struggles for a Toehold in Korea." Harvard Asia Quarterly 10(1): 25-30: https://web.archive.org/web/20080516063253/http://www.asiaquarterly.com/content/view/167/43/

This lack of research is being impressively addressed by In-Sung Kim Han, and she covers Medieval Islam in Korea in a conference paper at the 8th World Congress of Korean Studies: http://congress.aks.ac.kr/korean/files/2_1478844820.pdf (translations above are quoted from this paper)

For more on Muslim traders in Medieval Korea, see Seong Min Hong, "Commercial Relations between Korea and the Middle East in the Medieval Ages", papers presented at The 10th International Conference of KAMES, October 26-28, 2001, Korean Association of the Middle East Studies. Seoul: Korean Journal of the Middle East Studies. No. 22-2. 2002: http://hopia.net/hong/file/english/commercial.htm