Why does Raina call herself a "prosaic little coward"?

To be "prosaic" is to be concerned with everyday, mundane, practical considerations. It is the opposite of being poetic, romantic, or focused on exalted subjects. So Raina is being self-deprecating, and chiding herself for doubting that Sergius could live up to the romantic ideals from her favorite literary works and operas. She makes reference to cowardice because she had 

"…an uneasy fear that [Sergius] might cut a poor figure there beside all those clever Russian...

To be "prosaic" is to be concerned with everyday, mundane, practical considerations. It is the opposite of being poetic, romantic, or focused on exalted subjects. So Raina is being self-deprecating, and chiding herself for doubting that Sergius could live up to the romantic ideals from her favorite literary works and operas. She makes reference to cowardice because she had 

"…an uneasy fear that [Sergius] might cut a poor figure there beside all those clever Russian officers."

In Raina's view -- based on her romantic stories from literature and opera -- there are clearly defined sex roles. Great men do heroic deeds. Women aren't called upon to show as much physical courage as the men do -- they don't have to risk death in battle. But they are supposed to demonstrate their mettle by showing unwavering faith in the heroism of their men folk. They don't question whether the whole thing is a misguided delusion, or fear that their men are going to fail or die. They aren't troubled by the anxious thought that their men are just regular people with regular needs and weaknesses.

But that's just what Raina had been doing. As she confesses to her mother:

Well, it came into my head...that perhaps we only had our heroic ideas because we are so fond of reading Byron and Pushkin, and because we were so delighted with the opera that season at Bucharest... [Remorsefully] Only think, mother: I doubted him: I wondered whether all his heroic qualities and his soldiership might not prove mere imagination when he went into a real battle.

Her mother agrees: Raina shouldn't have doubted him. But Raina feels happy relief to think that her doubts have been proven wrong, and is able to laugh about her former status as a "prosaic little coward."