groupby
+ map
u = df1.groupby("USER")["ID"].agg(list)
df1["MULTI_IDS"] = df1["USER"].map(u[u.str.len().ge(2)])
USER ID MULTI_IDS
0 BOB 1 [1, 7]
1 STEVE 2 [2, 5, 6]
2 PAUL 3 NaN
3 KEITH 4 NaN
4 STEVE 5 [2, 5, 6]
5 STEVE 6 [2, 5, 6]
6 BOB 7 [1, 7]