I've noticed that nowhere I can find can give me a definitive answer to the question above. I first wondered about this when I noticed that you never had to state the type of variable in QBasic when declaring them, although you could add a suffix to the name of a variable to make sure it was of a particular type.
Also, as some dialects of BASIC are interpreted and others compiled, does this affect the typing system?
There are so many flavors of BASIC, some only historical and some still in use, that it's impossible to give one true answer.
Some of the old BASICs (the line numbered BASICs) had two data types: String, or Integer. The original BASIC that shipped with Apple-II computers was an "Integer BASIC." Later BASICs introduced Floating Point, which was often single precision FP. The BASIC that shipped with the TI-99/4a was an example of an early-80's floating point BASIC. "Way back when", you would make a string literal with quotes, and a string variable with a $ sigil following the identifier name. Variables that didn't have the $ sigil would usually default to the type of numeric variable that the given flavor of basic supported (Integer or Floating Point). GWBasic, for example, would default to floating point unless you specified the % sigil, which meant "Integer". TI Extended Basic didn't have an integer type, but the floating point numeric type had something like 15 significant digits, if I recall (floating point math errors not withstanding).
These early basics were essentially statically typed, though the distinction was far less useful than in more powerful languages. The choices for data types were few: String, Number (sometimes Int, sometimes FP), and sometimes with the ability to specify whether a number as Int or FP. Behind the scenes some even freely converted between ints and floating point as necessary. Often such behind the scenes conversions were not well documented.
But that was the state of affairs in the '80s, when everyone with a home computer was a hobbiest, and standards were loose. Every hardware manufacturer seemed to have their own take on how BASIC should work.
More modern BASICs are more powerful, and allow for tighter control over variable types (when needed).