Where your crime points have numeric fields which are amenable to simple statistics (mean, max, min, sum and median), this is straightforward. In QGIS go Vector->Data Management->Join Attributes by Location and set your target layer to the polygons and the 'join vector layer' to your points, then choose the summary statistics (second option). In ArcGIS, use Spatial Join and in the field mappings set the statistics options.
Text fields and numeric fields that represent classes or types of objects (i.e. where statistics would be meaningless) pose a bit more of a problem. If this is what you need to get, I would write a script using Python or ArcPy and have an iterative process that cycles through the polygons, selects points by the current polygon and then iterates over my selection to build a comma delimited text field. Alternatively, you could have a number of fields that are perhaps 'bins' to count crimes by type or something.
However, a better approach might be to write your script the other way round and cycle through the points, selecting polygons that contain them and append just the polygon IDs to a delimited text field. You then have a table of all the crimes which has a field listing all the nearby properties and you have preserved all the crime data without any summary. You can then analyse this in all sorts of ways to understand the associations of crimes and properties by selecting all crimes associated with a single property and perhaps creating a pivot table.
Best Answer
If you have a column in the attribute table that stores the diameter values, you can use that column to feed the buffer tool as follows:
Data Defined Overrideselect the column that stores the Diameter values.Here is the output: