As a premable, you need to understand the allied attitude in the context of the bombing of Warsaw, Rotterdam, London, Coventry etc. as well as the German naval raids on British coastal towns, and Zepplin raids on London in the previous war.

So in the context, the allies, and particulalry the British, were a lot less squeamish about the idea of inflicting civillian casualties than we might be today.

For the RAF: Started the war with the conceit that daylight bombers would hit individual target buildings. When this proved too costly, switched to night bombing with no electronic navigation. The Butt Report made it clear that astral navigation and dead reckoning was so poor that some bombers couldnt even find the right country, let alone a particular factory within a particular city within that country, the RAF switched to the practise of 'Area Bombing' This doctrine was partly based on the observation that civillians really didnt like it when they were dehoused by bombing and partly on the fact that greater accuracy was simply not achievable given the state of navigation technology at the time. Put simply, the RAF aimed to bomb a city with the express intent of dehousing the workforce.

Martin Middlebrook in The Nuremberg Raid gives a stark example of this. Nuremberg had a medieval centre with modern suburbs to the north, factories to the south, and extensive Nazi party infrastructure south of the factories. At this point in the war, the RAF would utilise a bomber stream with pathfinders marking the target with flares. This lead to a phenomenon called 'creepback' where succesive bombers would (understandably) tend to drop their bombs oh-so-slightly early, resulting in a spread of bomb-fall leading back from the aiming point in the direction from which the bomber stream arrived. Thus, if a bomber stream was arriving from the west, and you wanted to hit the centre of a city, the markers would be to the east of the centre. Make sense?

So, given that the bomber stream on the Nuremberg raid would be coming from the North, where would you place the target markers?

Now if you guessed at the extreme south of the town, you would be wrong. The markers were to be place in the centre of the city, meaning that the bomb-fall would cover the centre and the suburbs and leave the industrial area and Nazi stuff untouched

For the USAAF: Bombed in daylight, again with the conceit of hitting a specific target. This was all very well in theory, but the USAAF flew in rigid formations scientifically designed to optimise the combined defensive firepower of the bombers. Such formations could be very large indeed, depending on the number of bombers involved. Whats more, all of the bombers would drop as the lead bomber dropped.

You dont have to be a rocket scientist to figure out that this meant many square miles of a city would end up being bombed, rather than just a specific target.

So in summary, to the question of how concerned they were about civillian casualties, the answer is not at all. This was after all a total war. In fact it was only once the war was effectively won that people in the west started to question the morality bombing civillians.

Martin Middlebrook - The Nuremburg Raid

Williamson Murray - Luftwaffe: Strategy for Defeat

R V Jones - Most Secret War

Ralph Barker - The Thousand Plan

Just to add, operationally /u/jonewer is absolutely correct but politically leaders (especially Churchill) distanced themselves from the bombing of cities like Dresden / Nuremberg, even though they had permitted the practice.