In 3x17 - Visionary, the Romulan delegate mentions that the Dominion are the biggest threat to the Alpha quadrant - how so? The Romulans are certainly also aware of the Borg, and at least until this episode the Dominion merely managed to send a few Jem'Hadar through the Wormhole. Why aren't the Borg considered a bigger threat?
I think the best way to answer this is to look at all the major powers who either A. are an Alpha/"Beta on the border" Power, B. are capable of easily traveling to that part of the Alpha quadrant, or C. both; and comparing them. Note that this answer is very TNG era oriented, and is abridged in some ways
The Romulans: The Romulans are a major power, but they're not exactly a major threat. The precursor to the Federation, the Coalition of Planets, was able to force the Romulans into a stalemate, despite only having around 13 federation-precursor powers. The 24th century Federation is more than 5 times that size. Should the need arise, the Federation alone could defeat the Romulans. Furthermore, they also had the reunification movement, which was an internal issue that hindered their government's effectiveness. http://en.memory-alpha.wikia.com/wiki/Coalition_of_Planets
The Klingons: The Klingons were stalemated by the Federation in the 23rd century, and had dealt with a major civil war just a few years previously. They had been dealing with significant government corruption "Ezri Dax covered this in more detail than is needed here".
The Cardassians: The Cardassians were a major power, but at this point in DS9, they and the Federation were having issues just dealing with the Maquis, a group of colonists. On top of that, they were having domestic issues at the time as well, with the movement for a civilian government growing. The Cardassians had too many problems to be the top power.
The Federation: The Federation was one of the biggest powers, but they had the same problems with the Maquis that the Cardassians did. Furthermore, they had a lot of political division. For examples:
- Seven of Nine said "Officially, the Ktarians were with the Federation, but they sympathized with the Maquis." in "The Voyager Conspiracy".
- Many Federation colonies had been given to the Cardassians, which ticked off the citizens, and also caused disputes in the general population.
- In “The Drumhead” with Next Gen, we saw that there's actually a fair amount of McCarthyism in the Federation.
- We saw in an episode of Next Gen that at least one person had defected from the Federation to the Romulans, suggesting some internal political issues.
- There was a movement on Vulcan to align with the Romulans, which we saw in season 4 of Star Trek Enterprise. This could disrupt the membership of a core member of the Federation.
The Ferengi: The Ferengi were an economic power, not a military power.
The Borg: The Borg may be powerful, but they had only attacked with 1 or 2 vessels at this time, depending on how you count the ones left over that the NX01 encountered.
The Dominion: At this point in the series, the full extent of the Dominion wasn't known "unknown threats are greater threats". The dominion had the Changelings, which are a major political threat. The Dominion could destroy a Galaxy class with just a few fighters in minutes. The Dominion had a partial ability to see through Romulan cloaking devices "which would make the Romulans perceive them as a bigger threat." Also, we saw the Dominion defeat the Defiant in combat, which was a vessel designed to enter a fight with a Borg Cube. If the dominion can defeat vessels designed to defeat the Borg, it's a threat.
In summary, the Dominion was a bigger threat because it wasn't as well known as the other powers, could see through some Romulan cloaks, could defeat extremely powerful vessels, had the changelings, and the surrounding powers were too plagued by internal matters to be a threat.
In the Homefront Episode/Paradise Lost arc,
The dominion effectively infiltrated Star Fleet with very few operatives, and effectively crippled the Federation militarily without a single battle. If several hundred operatives (shapeshifters) had participated, likely the Federation would have remained in their control, as suggested in those episodes. The nature of this threat removes the military measures as a viable solution rendering intelligence and espionage as the only viable solution. Given the shapeshifters are shown to be vastly more able at espionage than the solids, Alpha Quadrant ability to stop that threat is effectively zero.
From the Romulan perspective this would represent a loss of sovereignty and would be understandably viewed as a dominant threat.