I think you really have to ask as to what you mean by "Civil War era".
There are dozens upon dozens of forts that remain to this day, many under the direction of the NPS, that have ACW ties. Many of those have been mentioned in this thread already and having visited most of them, they are fantastic examples of engineering in the US in the mid 1800s. The Third-System forts are obviously the best constructed and most intriguing as our resident expert,
@jrweaver, will testify.
If you mean by the Civil War alone, obviously no major masonry/stone structure was built in those times. At least not as a stand alone fortification.
That said, I've gotta go with Fort Duffield in Kentucky in terms of preservation of the ones I've visited. This fort's earthworks are in extremely good shape after all this time. The layout of the fort is clear and concise. I was very impressed upon my first visit a year or so ago. It doesn't have any sort of battle history but if you're a fan of engineering and fortifications. This is one you don't want to miss.
If you include masonry forts of the Third System - what a lot of people refer to as Civil War forts, as many saw action in the war - then that opens up a whole new set of forts.
The size of a fort makes a difference. If you talk about a smaller fort, it is easier for the whole fort to be well preserved. When you get to the very large forts, it is more likely that portions of the fort will be well preserved, but other areas will not. Also complicating things are the concrete fortifications of the Endicott Period, which were often built in or on Third System forts. Does the presence of an Endicott Battery mean the Third System fort is not well preserved? In many cases, probably the most dramatic being Fort Taylor on Key West, the Third System fort was greatly modified to accommodate the Endicott batteries. At Fort Taylor, the upper two tiers were removed!
With that in mind, I'll give a rundown of my opinion of the condition of the Third System. I'll do this in the order the Army always used in talking about the forts, northeast to south to northwest - a clockwise sweep.
Fort Knox, ME, very well preserved and well interpreted
Fort Popham, ME, not completed but what was built is well preserved. Reasonable signs to guide the visitor.
Fort Gorges, ME, well preserved with interpretive work and stabilization ongoing.
Fort Scammel, ME, was well preserved but now being turned into a luxury camp.
Fort Preble, ME, not completed, what was built remains. Some signs to guide the visitor.
Fort McClary, ME, not completed, but what was built is well preserved and nicely interpreted.
Fort Constitution, NH, not completed, some signage, more interpretation in process.
Fort Warren, MA, modified for Endicott, but most of the old fort in good shape. Some interpretation.
Fort Independence, MA, well preserved and decent interpretation.
Fort Taber, MA, well preserved but not accessible to the public. Nice museum near the fort has some interpretation.
Fort Adams, RI, very well preserved and interpreted. Very large fort with some areas not restored. Outworks are magnificent.
Fort Trumbull, CT, very well preserved and interpretation from previous management is very good.
Fort Schuyler, NY, adaptive reuse as SUNY Maritime Academy. Well cared for, but adapted for classroom, etc., use.
Fort Totten, NY, not completed, but what was built is in good condition. Well cared for now that it is a NYC park.
Fort Tomkins, NY, well preserved but access is limited. No interpretation except tours.
Fort Richmond, NY, extremely well preserved but access is limited to tours.
Fort Hamilton, NY, only a portion of the fort and a fantastic caponier remain. Well interpreted in caponier museum.
Fort Hancock, NJ, almost nothing remains of the original fort.
Fort Delaware, DE, very well preserved and well interpreted.
Fort Carroll, MD, neglected but still intact, no access or interpretation.
Fort Monroe, VA, very well preserved and well interpreted. This is a very large fort, so some areas not interpreted.
Fort Calhoun (Wool), VA, closed to the public and sand added within fort for use as bird sanctuary. At-risk historic site!
Fort Macon, NC, very well preserved and well interpreted.
Fort Caswell, NC, much destroyed and modified, but part of the old fort remains. Reasonable interpretation.
Fort Sumter, SC, most of fort destroyed between Civil War siege and Endicott construction. Reasonable interpretation.
Fort Pulaski, GA, very well preserved and very well interpreted.
Fort Clinch, FL, very well preserved and very well interpreted.
Fort Taylor, FL, most of fort destroyed during Endicott construction, but one tier of casemates remains. Well interpreted.
Fort Jefferson, FL, fort is in remarkable condition considering the hurricanes it has withstood. Well interpreted.
Fort Pickens, FL, much of the fort destroyed by accidental explosion and Endicott construction, but well interpreted.
Fort Barrancas, FL, very well preserved and very well interpreted.
Advanced Redoubt, FL, very well preserved. Access limited to tours.
Fort McRee, FL, nothing left of old fort.
Fort Morgan, AL, very well preserved and well interpreted.
Fort Gaines, AL, very well preserved, but modified for handicap accessibility. Quite well interpreted.
Fort Massachusetts, MS, well preserved, some interpretation.
Fort Pike, LA, structural and neglect issues. Very limited access.
Fort Macomb, LA, neglect issues. Very limited access.
Fort Jackson, LA, quite well preserved, but some Endicott modification. Limited access at this time.
Fort Livingston, LA, about half of the old fort remains due to wave action. No interpretation.
Fort Point, CA, most completely preserved and interpreted fort of the Third System.
Fort Alcatraz, CA, little portions of the original fort remain, but are well interpreted.
These are my opinions based on visits over the past two years.