Our solution is based on a DDL trigger. You can read more about it in BOL.

Create table [dbo].[DDLEventLog] where logged events will be stored. Create trigger [trg_DatabaseDDLeventer] which will fire on all database level DDL events ( DDL_DATABASE_LEVEL_EVENTS) and store event detail in XML format to [dbo].[DDLEventLog] table.

-- Create events table
CREATE TABLE [dbo].[DDLEventLog]
(
    [Id] INT IDENTITY(1,1) PRIMARY KEY,
    [EventDate] DATETIME,
    [Originator] NVARCHAR(256),
    [EventXML] XML
)
GO

-- Create trigger
SET ANSI_NULLS, QUOTED_IDENTIFIER ON
GO

CREATE TRIGGER [trg_DatabaseDDLeventer] ON DATABASE
FOR DDL_DATABASE_LEVEL_EVENTS AS
BEGIN
    
	INSERT INTO [dbo].[DDLEventLog] 
        ([EventDate], [Originator], [EventXML])
    VALUES
        (GETDATE(), USER_NAME(), EVENTDATA())

END
GO

We will run the DDL statement CREATE TABLE to test how it works.

-- Create sample DDL event to test trigger
CREATE TABLE [dbo].[TestTable] ([Id] int)
GO

-- Get results
SELECT * FROM [DDLEventLog]
GO

If we will click on the link in EventXML column we can observe message details and identify who executed which DDL statement:

<EVENT_INSTANCE>
  <EventType>CREATE_TABLE</EventType>
  <PostTime>2010-07-11T10:55:04.710</PostTime>
  <SPID>51</SPID>
  <ServerName>TESTSERVER</ServerName>
  <LoginName>TESTSERVER\Admin</LoginName>
  <UserName>dbo</UserName>
  <DatabaseName>master</DatabaseName>
  <SchemaName>dbo</SchemaName>
  <ObjectName>TestTable</ObjectName>
  <ObjectType>TABLE</ObjectType>
  <TSQLCommand>
    <SetOptions ANSI_NULLS="ON" ANSI_NULL_DEFAULT="ON" ANSI_PADDING="ON" QUOTED_IDENTIFIER="ON" ENCRYPTED="FALSE" />
    <CommandText>CREATE TABLE dbo.TestTable (Id int)</CommandText>
  </TSQLCommand>
</EVENT_INSTANCE>

If you would like to implement a more robust and security compliant solution then please review SQL Server Audit functionality for more details.