@app.route('/api/users/<int:user_id>', methods=['PUT'])
def put_user(user_id):
    user = {
        'id': user_id,
        'login': request.json['login'],
        'password': request.json['password'],
    }
    cursor.execute('UPDATE USERS SET login=?, password=? WHERE id=?', (user['login'], user['password'], user['id']))
    connection.commit()
    return jsonify(user), 200

{
    "id": 4,
    "login": "bbbb",
    "password": "aaaa"
}

[
     {
         "id": 1,
         "login": "123",
         "password": "1231231"
     },
     {
         "id": 2,
         "login": "login",
         "password": "sfsdfsd"
     },
     {
         "id": 3,
         "login": "4444",
         "password": "sass"
     },
     {
         "id": 4,
         "login": "ffff",
         "password": "aaaa"
     }
]