Yes, you can re-use the connection: you do not need to create a new one. You can also re-use the Reader, and the command.
More importantly, do not do "SELECT *" and then rely on the ordinal position of fields: they may not stay that way. Reader[2] works, but if you alter your table definition, it may not. Instead use the string form, or better still select only those fields you are actually going to use: this will also save bandwith in communicating with the database server.
You could even just use a DataAdapter and set the DataSource property of the combo box to the selected information directly:
using (MySqlConnection connection = new MySqlConnection(MyConString))
{
connection.Open();
using (MySqlCommand command = new MySqlCommand("SELECT course FROM course_master", connection))
{
MySqlDataAdapter da = new MySqlDataAdapter(command);
DataTable dt = new DataTable();
da.Fill(dt);
cmbo_course.DataSource = dt;
}
}