How to create a smooth black and white texture without jaggies?

Here's another option using Rasterize.

SeedRandom[1234];
img = Binarize[
   Blur[ListDensityPlot[randomTiles, InterpolationOrder -> 0, 
     Frame -> False, PlotRangePadding -> 0, 
     ImageSize -> {1600, 1600}], 40]];
rimg = Rasterize[img, RasterSize -> 400, ImageSize -> 400]
ImageDimensions[rimg]

• Use ColorFunction -> Graylevel in ListDensityPlot to start from a black and white image, avoiding Binarize altogether.
• ask for a much higher number of pixels in your image with ImageSize; that alone causes considerable smoothing when you downsize the image;
• for further effect, apply e.g. MedianFilter with an appropriate parameter.

SeedRandom[1234]
randomTiles = 
  Table[{RandomReal[], RandomReal[], RandomInteger[{0, 1}]}, {n, 1, 1000}];


MedianFilter[#, 5] &@
 ListDensityPlot[
   randomTiles,
   InterpolationOrder -> 0, Frame -> False,
   PlotRangePadding -> 0, ImageSize -> {1000, 1000},
   ColorFunction -> GrayLevel
 ]