Query with multiple values in a column
The best solution would be to normalize your schema. You should have a separate table with one row for each child, instead of a comma-delimited list. Then you can join with this table to find parent with a specific child. See @themite's answer for an example of this.
But if you can't do that for some reason, you can use FIND_IN_SET:
WHERE FIND_IN_SET('Alex', children)