We present high spatial resolution (similar to 1.0('')) imaging of Br gamma, H-2 1-0 S(1) 2.121 mu m and [Fe II] 1.64 mu m line emission and broad-band J, H and K continuum emission of the nuclear (20('') = 1.6 kpc diameter) region of the starburst galaxy NGC 1808. These data, combined with a 3.6 cm radio continuum map, are used to study the morphology and extinction of the starburst region, the nature of the stellar population and star formation, and the possibility of hidden Seyfert activity in NGC 1808. The broad-band continuum emission is smoothly distributed throughout the circumnuclear region and peaks strongly on the nucleus. There is no strong morphological change with wavelength. Most of the continuum emission is probably produced by the evolved bulge stellar population, and not by hot dust, red giants or red supergiants, as in most other starburst galaxies. The line emission is dominated by the nucleus, but is extended along the major axis of the galaxy by similar to 18('') (similar to 1.4 kpc) for all emission lines. The circumnuclear emission arises from several distinct regions, forming a ring-like structure. Both Br gamma and [Fe II] are well correlated with the radio emission. There are, however, differences in the detailed morphology, that differentiate between hot spots dominated by supernova remnants and HII regions. The H-2 emission is slightly better spatially correlated with Br gamma than with [Fe II] emission but no firm conclusion about the excitation mechanism of the H-2 is warranted. The Br gamma emission effectively isolates the star forming activity, whereas radio and [Fe II] emission is affected by individual supernova explosions, and the optical H alpha emission peaks are only seen in directions of low extinction. From comparison of Br gamma and H alpha fluxes we derive extinctions between A(V) of 3 and 5 towards the hot spots. From analysis of the extinction-corrected line and continuum emission luminosities using an evolutionary starburst model, we derive for each hot spot star forming rate 0.1-0.6 M(.) yr(-1), and supernova rate 0.4-11 x 10(-3) yr(-1). We estimate the age of the current burst to be between 8 and 17 Myr in the circumnuclear region, and similar to 40 Myr in the nucleus. The circumnuclear starburst hot spots lie in the area of HII regions and starburst galaxies in the [Fe II]/Br gamma vs. H-2/Br gamma diagram, whereas the nucleus has line ratios similar to Seyfert nuclei. The multiwavelength evidence for and against hidden Seyfert activity in NGC 1808, and implications for the evolution of starburst galaxies are discussed.