Add ID found in list to new column in pandas dataframe

using merge and concat whilst grouping by your index to return all the matches.

bad_ids = [15533, 876544, 36789, 11111]

df2 = pd.concat(
    [
        df,
        pd.merge(
            df["Found_IDs"].explode().reset_index(),
            pd.Series(bad_ids, name="bad_ids"),
            left_on="Found_IDs",
            right_on="bad_ids",
            how="inner",
        )
        .groupby("index")
        .agg(bad_ids=("bad_ids", list)),
    ],
    axis=1,
).fillna(False)
print(df2)


      ID                   Found_IDs          bad_ids
0  12345        [15443, 15533, 3433]          [15533]
1  15533  [2234, 16608, 12002, 7654]            False
2   6789      [43322, 876544, 36789]  [876544, 36789]

You can apply and use np.any:

df['bad_id'] = df['Found_IDs'].apply(lambda x: np.any([c in x for c in bad_ids]))

This return the bool if exist a bad_id in Found_IDs, if you want to retrieve this bad_ids:

df['bad_id'] = df['Found_IDs'].apply(lambda x: [*filter(lambda x: c in x, bad_ids)])

This will return a list of the bad_ids at found_ids, if there is 0 it returns []

Using np.intersect1d to get the intersect of the two lists:

df['bad_id'] = df['Found_IDs'].apply(lambda x: np.intersect1d(x, bad_ids))

      ID                   Found_IDs    bad_id
0  12345        [15443, 15533, 3433]   [15533]
1  15533  [2234, 16608, 12002, 7654]        []
2   6789      [43322, 876544, 36789]  [876544]

Or with just vanilla python using intersect of sets:

bad_ids_set = set(bad_ids)
df['Found_IDs'].apply(lambda x: list(set(x) & bad_ids_set))

If want test all values of lists in Found_IDs column by all values of bad_ids use:

bad_ids = [15533, 876544]

df['bad_id'] = [any(c in l for c in bad_ids) for l  in df['Found_IDs']]
print (df)
      ID                   Found_IDs  bad_id
0  12345        [15443, 15533, 3433]    True
1  15533  [2234, 16608, 12002, 7654]   False
2   6789      [43322, 876544, 36789]    True

If want all match:

df['bad_id'] = [[c for c in bad_ids if c in l] for l  in df['Found_IDs']]
print (df)
      ID                   Found_IDs    bad_id
0  12345        [15443, 15533, 3433]   [15533]
1  15533  [2234, 16608, 12002, 7654]        []
2   6789      [43322, 876544, 36789]  [876544]

And for first match, if empty list is set False, possible solution, but not recommended mixing boolean and numbers:

df['bad_id'] = [next(iter([c for c in bad_ids if c in l]), False) for l  in df['Found_IDs']]
print (df)
      ID                   Found_IDs  bad_id
0  12345        [15443, 15533, 3433]   15533
1  15533  [2234, 16608, 12002, 7654]   False
2   6789      [43322, 876544, 36789]  876544

Solution with sets:

df['bad_id'] = df['Found_IDs'].map(set(bad_ids).intersection)
print (df)

      ID                   Found_IDs    bad_id
0  12345        [15443, 15533, 3433]   {15533}
1  15533  [2234, 16608, 12002, 7654]        {}
2   6789      [43322, 876544, 36789]  {876544}

And also similar with list comprehension:

df['bad_id'] = [list(set(bad_ids).intersection(l)) for l  in df['Found_IDs']]
print (df)
      ID                   Found_IDs    bad_id
0  12345        [15443, 15533, 3433]   [15533]
1  15533  [2234, 16608, 12002, 7654]        []
2   6789      [43322, 876544, 36789]  [876544]