you could dialyse against TE or precipitate using salt ethanol or just add an equivalent amount of HCL to just neutralise the NaOH leaving your DNA in NAcl salt or even run the sample through a desalting sephadex column when the dna comes off the column first leaving the salt in the sephadex matrix. Assuming that you washed the Dyna beads well then the ssDNA should already be very pure
PicoGreen which is very sensitive and specific for dsDNA. I used to measure DNA polymerase activity using PicoGreen with ssDNA as substrate with low background signal from ssDNA. Ideally you can use a calibration control with known ss/ds ratio or measure the 260nm absorption of total DNA versus PicoGreen.
How do I separate ssDNA and dsDNA at 70 °C without destroying either one of them? - ResearchGate. Available from: https://www.researchgate.net/post/How_do_I_separate_ssDNA_and_dsDNA_at_70_C_without_destroying_either_one_of_them [accessed Jan 29, 2017].
Finally was able to extract ssDNA >4kb from dsDNA plasmid by double digesting with nicking enzymes. The digested plasmid was directly loaded in denaturing gel loading buffer and electrophoresed on non denaturing gel with ethidium bromide. The correct size band was gel extracted.
You can design your own insert with nicking sites or clone it in a plasmid from commercial source (http://www.biodynamics.co.jp/images/news/ds610.pdf) which also provides denaturing loading buffer.