You should try to avoid drilling into brick if at all possible. You should drill into your horizontal mortar joints instead. That way, if you ever remove that gate, you can repack the holes in the mortar with new mortar for an invisible repair.
Even 1/4 inch diameter plastic anchors should be plenty strong enough to do what you're wanting. You can also use lead anchors and #12 screws for greater strength. I personally think that plastic anchors and lead anchors are one of the best ways to fasten anything to brick because they can be removed fairly easily by just drilling the plastic or lead out of the hole.
If you have a 4 inch wide board and you want to fasten it vertically to a brick wall, I wouldn't use any more than one anchor per horizontal mortar joint. But, I'd alternate the side of the board the anchor is on so that the board doesn't wobble at all. I'd probably put one anchor every 2 horizontal mortar joints (maybe 3) on alternating sides of the board.
Where I live, the kinds of plastic anchors you see below are about the most popular. They're strong enough for what you're wanting to do. They come in 3/16, 1/4 and 5/16 inch sizes for #8, #10 and #!2 size screws respectively. If it were me, I'd use the 1/4 inch size plastic anchors with #10 screws.
For more strength, you can use lead anchors with a #10 or #12 screw:
Also, to drill into brick mortar, you really should use a hammer drill and a masonary bit. These kinds of drill bits don't "cut" a hole the way wood and metal drilling bits do. Instead, they batter a hole in the masonary. So, you need a drill that will provide a simultaneous rotating and battering action, not just rotation. Such drills are called "hammer drills". Percussion type hammer drills are relatively inexpensive, or you could probably borrow one from any of the homeowners you work with or people you know who do home renovations and repairs.
The way I would proceed with installing this board would be to put it in it's desired location on the wall and mark the locations of the horizontal mortar joints on the board.
Then, mark the screw hole locations on the board and drill 1/8 inch diameter holes through the board.
Now, put the board in it's desired location again, and using only two or three holes in the board as a guide, use a 1/8 inch MASONARY BIT and hammer drill to drill 1/8 inch diameter holes into the horizontal mortar joints at those two or three locations.
Now take the board away and use a 1/4 inch masonary bit to enlarge those two or three holes in the mortar joints to 1/4 inch in diameter. Wrap a piece of thin masking tape around the 1/4 inch drill bit as a depth gauge so that you drill the holes to the right depth.
Use a vaccuum cleaner and drinking straw to clean the dust out of the holes, and hammer the plastic anchors into the two or three holes with a small hammer.
Mount the board to the brick wall with screws.
Now drill through all the remaining holes with the 1/8 inch masonary bit, remove the board, enlarge all the holes and drill them to the correct depth, vaccuum out all the dust, hammer in all the anchors and mount the board with screws in every hole.