London is a big place - which parts of it do you want to get to?
Of the 2 you list, Biggin Hill has a much higher level of aviation services than Elstree, but if you are arriving VFR, that should not make a difference. Biggin Hill also has a longer runway, but again, that may not be very useful.
Landing and parking fees are much lower at Elstree, but in the big scheme of things hardly likely to make a material difference. Getting in and out of Elstree will be quicker than Biggin Hill because it is smaller...
Neither Biggin Hill nor Elstree (nor any other GA airfield I know of...) have very easy, door to door links to central London.
If you plan to take a taxi, Elstree would be quicker than Biggin Hill; E.g. Google lists Elstree as 14 miles, 40 minutes, Biggin Hill as 17 miles, 52 minutes - both of those to Charing CRoss - and you may need to double that time depending on traffic.
Knowing London traffic, disregarding the cost of a taxi all the way, the quickest way is likely to be public transport. A very short taxi ride from Elstree to the local Thameslink suburban train service will get you into e.g. Kings Cross/St.Pancras or the City itself in about 20 minutes.
Alternatively, Elstree is close to the end of the Northern line and the Jubilee Line which serve other parts of Central London, although the underground is typically slower than the trains.
Not quite sure when SGC was last at Elstree w.r.t. it being 'run down'... I have seen both far worse and far nicer airfields in the UK. They have invested in it over the last couple of years and the new café is actually quite good. But it does fall somewhere in between a 'biz jet FBO' and a 'grassroots GA field'... I guess you get what you pay for.
You can also look at Stapleford (Central line into London). Denham and Fairoaks are further out but may also work for you depending on where you want to get to...
B.