분 is the polite word for 'person' and 계시다 is the honorific of 있다 ('there is'), so a modifying clause plus 분 계시다 asks whether anyone fits the description. Nearly every line here is aimed at an audience or a group of guests: 감기 걸리신 분 계신가요? (has anyone caught a cold?), 탈락하신 분 계신가요? (is anyone out?). The honorific is chosen because the people being asked about are the listeners themselves — with 사람 있어요? the same question would sound blunt.