While you could put silicone (or acrylic) caulk between the glass and wood, there's no need to.
This very much appears to be an internal door. There's no need to prevent air or water transfer from one side to the other. Even if it's an external door, windows sashes like this are not normally installed that way.
Any caulking will have an adhesive effect that will make it difficult to remove the pane in the future if any repairs (to the leading between the panes, replacement of a damaged pane, or the wood in the frame) should become necessary.
Your 1/2" x 1" wooden slats are usually called a "stop", and they're to hold one side of the window in place.
On the other side, you indicate that there is some sort of a "clamp" holding the window in place. Usually the other side is held in by several glazier's points*, I hope this is what you mean by a "clamp".
The glass is then trimmed out using glazier's compound on both sides. It is a non-hardening putty, similar to plumber's putty, that comes in a can or tub. You roll it in your hands to warm & soften it, then force it (by hand) into a fillet between the glass and wood, then tool it (with a tool or by hand) to a smooth, pleasing appearance. When done properly, it will make a weather tight seal that will hold for a decade or more for external use. Of course, for your interior transom, you only need it for appearance, as it's unlikely to rain on either side of the door, nor should you be worried about an air-tight seal there, either.
*Glazier's points are small, flat metal pins that are placed on the glass and pushed into the frame up to their built-in stops. The part that remains visible holds the glass, while the part that goes into the wood holds it tight to the wood. The visible part is then covered by the glazing compound. When installed properly, they will prevent any rattling. Oddly, my search for "glazier points" led me to a number of places to buy them, and a couple of instructions on how to install them, but not to anything that actually described what they are.